# questions

tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
title:
the-torque-of-a-force-5-i-3-j-7-k-about-the-origin-is-if-the-force-acts-on-a-particle-whose-position-vector-is-2-i-2-j-k-then-the-value-of-will-be-1

slug:
the-torque-of-a-force-5-i-3-j-7-k-about-the-origin-is-if-the-force-acts-on-a-particle-whose-position-vector-is-2-i-2-j-k-then-the-value-of-will-be-1-63400

question:
<div class="question">The torque of a force $5 \hat{i}+3 \hat{j}-7 \hat{k}$ about the origin is $\tau$. If the force acts on a particle whose position vector is $2 \hat{i}+2 \hat{j}+\hat{k}$, then the value of $\tau$ will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$11 \hat{\mathrm{i}}+19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-11 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}-16 \hat{\mathrm{k}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-17 \hat{\mathrm{i}}+19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$17 \hat{i}+9 \hat{j}+16 \hat{k}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-17 \hat{\mathrm{i}}+19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$</span> </div>

answer:
<div class="solution">Given that torque, $\vec{\tau}=5 \hat{i}+3 \hat{j}-7 \hat{k}$<br/>Position vector, $\overrightarrow{\mathrm{r}}=2 \hat{\mathrm{i}}++2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>$\begin{aligned} &amp; \vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}=\left|\begin{array}{ccc}\mathrm{i} &amp; \mathrm{j} &amp; \mathrm{k} \\ 2 &amp; 2 &amp; 1 \\ 5 &amp; 3 &amp; -7\end{array}\right| \\ &amp; =\hat{\mathrm{i}}(-14-3)-\hat{\mathrm{j}}(-14-5)+\hat{\mathrm{k}}(6-10) \\ &amp; =-17 \hat{\mathrm{i}}+19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}\end{aligned}$</div>


tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
title:
the-total-energy-of-an-electron-in-the-first-excited-state-of-hydrogen-is-about-3-4-ev-its-kinetic-energy-in-this-state-is

slug:
the-total-energy-of-an-electron-in-the-first-excited-state-of-hydrogen-is-about-3-4-ev-its-kinetic-energy-in-this-state-is-11782

question:
<div class="question">The total energy of an electron in the first excited state of hydrogen is about <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>eV</mi></math>. Its kinetic energy in this state is </div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>eV</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>.</mo><mn>8</mn><mo> </mo><mi>eV</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>8</mn><mo> </mo><mi>eV</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>eV</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>eV</mi></math></span> </div>

answer:
<div class="solution"><p>We have, Kinetic energy of electron,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo>=</mo><mfrac><mrow><mi>Z</mi><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mn>8</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><mo>·</mo><mi>r</mi></mrow></mfrac></math></p><p>also, Potential energy of electron,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac><mo>·</mo><mfrac><mrow><mi>Z</mi><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Total energy</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>K</mi><mo>+</mo><mi>U</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>Z</mi><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mn>8</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><mo>·</mo><mi>r</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>Z</mi><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><mo>·</mo><mi>r</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>Z</mi><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mn>8</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><mi>r</mi></mrow></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo>=</mo><mo>-</mo><mi>E</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mo>-</mo><mn>3</mn><mo>.</mo><mn>4</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>eV</mi></math></p></div>


title: the-total-number-of-4-digit-numbers-in-which-the-digits-are-in-descending-order-is
slug: the-total-number-of-4-digit-numbers-in-which-the-digits-are-in-descending-order-is-36255
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
question:
<div class="question">The total number of 4 -digit numbers in which the digits are in descending order, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }^{n} C_{4} \times 4 !$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">${ }^{10} \mathrm{C}_{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{10 !}{4 !}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }^{10} \mathrm{C}_{4}$</span> </div>

answer:
<div class="solution">Total number of arrangements of 10 digits $0,1,2, \ldots, 9$ bytaking 4 at atime $={ }^{10} \mathrm{C}_{4} \times 4 !$
<br/> <br/>We observe that in every arrangement of 4 selected digits there is just one arrangement in which the digits are in descending order.
<br/> <br/>$\therefore$ Required number of 4 -digit numbers.
<br/> <br/>$$
<br/> <br/>=\frac{{ }^{10} \mathrm{C}_{4} \times 4 !}{4 !}={ }^{10} \mathrm{C}_{4}
<br/> <br/>$$</div>


title: the-total-number-of-p-oh-bonds-for-pyrophosphoric-acid
slug: the-total-number-of-p-oh-bonds-for-pyrophosphoric-acid-49168
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
question:
<div class="question">The total number of $\mathrm{P}-\mathrm{OH}$ bonds for pyrophosphoric acid</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>4
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>5
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>6
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>8</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>4
<br/></span> </div>

answer:
<div class="solution">The structure of pyrophosphoric acid is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/6N37zROY_-PvgRSnSTWr7u_zDWKo5dnhVX499Ej9dzQ.original.fullsize.png"/><br/> <br/>There are $4 \mathrm{P}-\mathrm{OH}$ bonds, $1 \mathrm{P}-\mathrm{O}-\mathrm{P}$ bond and two $\mathrm{P}=\mathrm{O}$ bonds.</div>


title: the-transfer-rna-anticodon-for-the-messenger-rna-codon-g-c-a-is
slug: the-transfer-rna-anticodon-for-the-messenger-rna-codon-g-c-a-is-26729
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The transfer <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>RNA</mi></math> anticodon for the messenger <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>RNA</mi></math> codon <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">G</mi><mo>-</mo><mi mathvariant="normal">C</mi><mo>-</mo><mi mathvariant="normal">A</mi></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi><mo>-</mo><mi mathvariant="normal">G</mi><mo>-</mo><mi mathvariant="normal">U</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi><mo>-</mo><mi mathvariant="normal">C</mi><mo>-</mo><mi mathvariant="normal">U</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">U</mi><mo>-</mo><mi mathvariant="normal">C</mi><mo>-</mo><mi mathvariant="normal">C</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">G</mi><mo>-</mo><mi mathvariant="normal">U</mi><mo>-</mo><mi mathvariant="normal">C</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi><mo>-</mo><mi mathvariant="normal">G</mi><mo>-</mo><mi mathvariant="normal">U</mi></math></span> </div>

answer:
<div class="solution"><p>The transfer <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>RNA</mi></math> anticodon for messanger <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>RNA</mi></math> codon <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">G</mi><mo>-</mo><mi mathvariant="normal">C</mi><mo>-</mo><mi mathvariant="normal">A</mi></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi><mo>-</mo><mi mathvariant="normal">G</mi><mo>-</mo><mi mathvariant="normal">U</mi></math>, because in anticodon of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>RNA</mi></math>,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">G</mi></math> replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">G</mi></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">U</mi></math></p><p>Here, C(cytosine) forms triple bond with G (guanine) and A (adenine) forms double bond with U (uracil).</p></div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
title:
the-transverse-displacement-y-x-t-of-a-wave-on-a-string-is-given-by-y-x-t-e-a-x-2-b-t-2-2-ab-xt-this-represents-a

slug:
the-transverse-displacement-y-x-t-of-a-wave-on-a-string-is-given-by-y-x-t-e-a-x-2-b-t-2-2-ab-xt-this-represents-a-80789

question:
<div class="question">The transverse displacement $y(x, t)$ of a wave on a string is given by 
<br/>$y(x, t)=e^{-\left(a x^{2}+b t^{2}+2 \sqrt{a b}\right.}$ xt $)$
<br/>This represents a</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">wave moving in $-x$ direction, speed $\sqrt{\frac{b}{a}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">standing wave of frequency $\sqrt{b}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">standing wave of frequency $\frac{1}{\sqrt{b}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">wave moving in $+x$ direction, speed $\sqrt{\frac{a}{b}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">wave moving in $-x$ direction, speed $\sqrt{\frac{b}{a}}$</span> </div>

answer:
<div class="solution">$y(x, t)=e^{-\left(a x^{2}+b t^{2}+2 \sqrt{a b} x t\right)}=e^{-(\sqrt{a x}+\sqrt{b t})^{2}}$
<br/>It is a function of type $y=f(\omega t+k x)$ $\therefore y(x, t)$ represents wave travelling along $-x$ direction.
<br/>$\text { Speed of wave }=\frac{\omega}{k}=\frac{\sqrt{b}}{\sqrt{a}}=\sqrt{\frac{b}{a}}$</div>


tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">NOR</span> </div>
source_db: MarksBatch2_P3.db
title:
the-truth-table-given-below-correspond-to-the-logic-gate-begintabularccc-a-b-x-hline-0-0-1-1-0-0-0-1-0-1-1-0-endtabular

slug:
the-truth-table-given-below-correspond-to-the-logic-gate-begintabularccc-a-b-x-hline-0-0-1-1-0-0-0-1-0-1-1-0-endtabular-16465

question:
<div class="question">The truth table given below correspond to the logic gate<br/>\begin{tabular}{cc|c}<br/>$\mathrm{A}$ &amp; $\mathrm{B}$ &amp; $\mathrm{X}$ \\<br/>\hline 0 &amp; 0 &amp; 1 \\<br/>1 &amp; 0 &amp; 0 \\<br/>0 &amp; 1 &amp; 0 \\<br/>1 &amp; 1 &amp; 0<br/>\end{tabular}</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{OR}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">NOR</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">AND</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">NAND</span> </li> </ul>

answer:
<div class="solution">The given gate is a NOR gate. The output is high, when all inputs are low.<br/>Truth Table:<br/>\begin{tabular}{|c|c|c|}<br/>\hline $\mathbf{A}$ &amp; $\mathbf{B}$ &amp; $\mathbf{Y}$ \\<br/>\hline 0 &amp; 0 &amp; 1 \\<br/>\hline 1 &amp; 0 &amp; 0 \\<br/>\hline 0 &amp; 1 &amp; 0 \\<br/>\hline 1 &amp; 1 &amp; 0 \\<br/>\hline \multicolumn{2}{|c|} { Logic Symbols } \\<br/>\hline<br/>\end{tabular}<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/SAl1pjY7HKepM_U07OxoNd3okXgOIZ1vojjvHw6KR7A.original.fullsize.png"/><br/><br/>Boolean expression: $\overline{\mathrm{A}+\mathrm{B}}=\mathrm{Y}$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
title:
the-two-ends-of-a-rod-of-length-l-and-a-uniform-crosssectional-area-a-are-kept-at-two-temperatures-t-1-and-t-2-t-1-t-2-the-rate-of-heat-transfer-dt-dq

slug:
the-two-ends-of-a-rod-of-length-l-and-a-uniform-crosssectional-area-a-are-kept-at-two-temperatures-t-1-and-t-2-t-1-t-2-the-rate-of-heat-transfer-dt-dq-23497

question:
<div class="question">The two ends of a rod of length $L$ and a uniform crosssectional area $A$ are kept at two temperatures $T_{1}$ and $\mathrm{T}_{2}\left(\mathrm{~T}_{1}&gt;\mathrm{T}_{2}\right)$. The rate of heat transfer, $\frac{\mathrm{dQ}}{\mathrm{dt}}$ through the rod in a steady state is given by:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{k\left(T_{1}-T_{2}\right)}{L A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$k L A\left(T_{1}-T_{2}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{k A\left(T_{1}-T_{2}\right)}{L}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{k L\left(T_{1}-T_{2}\right)}{A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{k A\left(T_{1}-T_{2}\right)}{L}$</span> </div>

answer:
<div class="solution">$\frac{\mathrm{dQ}}{\mathrm{dt}}=\frac{\mathrm{kA}\left(\mathrm{T}_{1}-\mathrm{T}_{2}\right)}{\mathrm{L}}$ $\left[\left(\mathrm{T}_{1}-\mathrm{T}_{2}\right)\right.$ is the temperature difference $]$</div>


tags: ['Physics', 'Mathematics in Physics', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1,2,4</span> </div>
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P3.db
title:
the-two-vectors-a-and-b-are-drawn-from-a-common-point-and-c-a-b-then-angle-between-a-and-b-is-1-9-0-if-c-2-a-2-b-2-2-greater-than-9-0-if-c-2-a-2-b-2-3

slug:
the-two-vectors-a-and-b-are-drawn-from-a-common-point-and-c-a-b-then-angle-between-a-and-b-is-1-9-0-if-c-2-a-2-b-2-2-greater-than-9-0-if-c-2-a-2-b-2-3-50913

question:
<div class="question">The two vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ are drawn from a common point and $\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}},$ then angle<br/>between $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ is -<br/>(1) $90^{\circ}$ if $\mathrm{C}^{2}=\mathrm{A}^{2}+\mathrm{B}^{2}$<br/>(2) greater than $90^{\circ}$ if $\mathrm{C}^{2}&lt;\mathrm{A}^{2}+\mathrm{B}^{2}$<br/>(3) greater than $90^{\circ}$ if $\mathrm{C}^{2}&gt;\mathrm{A}^{2}+\mathrm{B}^{2}$<br/>(4) less than $90^{\circ}$ if $\mathrm{C}^{2}&gt;\mathrm{A}^{2}+\mathrm{B}^{2}$<br/>Correct options are -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1,2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1,2,3,4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2,3,4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">1,2,4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>


title: the-unit-of-rate-constant-for-reactions-of-second-order-is
slug: the-unit-of-rate-constant-for-reactions-of-second-order-is-99117
question: <div class="question">The unit of rate constant for reactions of second order is</div>
tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{L} \mathrm{mol}^{-1} \mathrm{~s}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{L}^{-1} \mathrm{~mol} \mathrm{~s}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{L} \mathrm{mol} \mathrm{s}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{s}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{L} \mathrm{mol}^{-1} \mathrm{~s}^{-1}$</span> </div>

answer:
<div class="solution">Unit of rate constant for second order reaction is $\mathrm{L} \mathrm{mol}^{-1} \mathrm{sec}^{-1}$</div>


title: the-unit-vector-perpendicular-to-the-vectors-6-i-2-j-3-k-and-3-i-6-j-2-k-is
slug: the-unit-vector-perpendicular-to-the-vectors-6-i-2-j-3-k-and-3-i-6-j-2-k-is-39549
tags: ['Mathematics', 'Vector Algebra', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
question:
<div class="question">The unit vector perpendicular to the vectors $6 \hat{i}+2 \hat{j}+3 \hat{k}$ and $3 \hat{i}-6 \hat{j}-2 \hat{k}$ is $-$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{2 \hat{i}-3 \hat{j}+6 \hat{k}}{7}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}}{7}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}}{7}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2 \hat{i}+3 \hat{j}+6 \hat{k}}{7}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}}{7}$</span> </div>

answer:
<div class="solution">Unit vector perpendicular to both the given vectors is, $\frac{(6 \hat{i}+2 \hat{j}+3 \hat{k}) \times(3 \hat{i}-6 \hat{j}-2 \hat{k})}{|(6 \hat{i}+2 \hat{j}+3 \hat{k}) \times(3 \hat{i}-6 \hat{j}-2 \hat{k})|}=\frac{2 \hat{i}+3 \hat{j}-6 \hat{k}}{7}$</div>


tags: ['Chemistry', 'Hydrocarbons', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
the-unsaturated-hydrocarbon-which-on-ozonolysis-gives-one-mole-each-of-formaldehyde-acetaldehyde-and-methyl-glyoxal-ch-3-cocho-is

slug:
the-unsaturated-hydrocarbon-which-on-ozonolysis-gives-one-mole-each-of-formaldehyde-acetaldehyde-and-methyl-glyoxal-ch-3-cocho-is-29154

question:
<div class="question">The unsaturated hydrocarbon which on ozonolysis gives one mole each of formaldehyde, acetaldehyde and methyl glyoxal $\left(\mathrm{CH}_{3} \mathrm{COCHO}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \mathrm{CH}_{3}-\mathrm{CH}=\mathrm{C}\left(\mathrm{CH}_{3}\right)-\mathrm{CH}_{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{CH}_{2}=\mathrm{CH}-\mathrm{CH}_{2}-\mathrm{CH}=\mathrm{CH}_{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\quad \mathrm{CH}_{2}=\mathrm{CH}-\mathrm{C}\left(\mathrm{CH}_{3}\right)=\mathrm{CH}-\mathrm{CH}_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\mathrm{CH}_{3}\right)_{2} \mathrm{C}=\mathrm{CH}-\mathrm{CH}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \mathrm{CH}_{2}=\mathrm{CH}-\mathrm{C}\left(\mathrm{CH}_{3}\right)=\mathrm{CH}-\mathrm{CH}_{3}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/1NTiYvc8QObIFEUxruwMqSMreF5mitcqq-BJrdWWNBs.original.fullsize.png"/></div>


tags: ['Physics', 'Mechanical Properties of Solids', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
the-upper-end-of-a-wire-of-diameter-12-mm-and-length-1-m-is-clamped-and-its-other-end-is-twisted-through-an-angle-of-3-0-the-angle-of-shear-is

slug:
the-upper-end-of-a-wire-of-diameter-12-mm-and-length-1-m-is-clamped-and-its-other-end-is-twisted-through-an-angle-of-3-0-the-angle-of-shear-is-26028

question:
<div class="question">The upper end of a wire of diameter $12 \mathrm{~mm}$ and length $1 \mathrm{~m}$ is clamped and its other end is twisted through an angle of $30^{\circ} .$ The angle of shear is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$18^{\circ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.18^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$36^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.36^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.18^{\circ}$</span> </div>

answer:
<div class="solution">$r \theta=\ell \phi \Rightarrow \phi=\frac{r \theta}{\ell}=\frac{6 \mathrm{~mm} \times 30^{\circ}}{1 \mathrm{~m}}=0.18^{\circ}$</div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
title:
the-upper-half-of-an-inclined-plane-with-inclination-is-perfectlysmooth-while-the-lower-halfis-rough-a-body-starting-from-rest-at-the-top-will-again-c

slug:
the-upper-half-of-an-inclined-plane-with-inclination-is-perfectlysmooth-while-the-lower-halfis-rough-a-body-starting-from-rest-at-the-top-will-again-c-34484

question:
<div class="question">The upper half of an inclined plane with inclination $\phi$ is perfectlysmooth while the lower halfis rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \cos \phi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \sin \phi$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\tan \phi$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2 \tan \phi$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \tan \phi$</span> </div>

answer:
<div class="solution">Acceleration of block while sliding down upper half $=\mathrm{g} \sin \phi$
<br/> <br/>retardation of block while sliding down lower half $=-(\mathrm{g} \sin \phi-\mu \mathrm{g} \cos \phi)$
<br/> <br/>For the block to come to rest at the bottom, acceleration in I half = retardation in II half.
<br/> <br/>$\mathrm{g} \sin \phi=-(\mathrm{g} \sin \phi-\mu \mathrm{g} \cos \phi)$
<br/> <br/>$\Rightarrow \mu=2 \tan \phi$
<br/> <br/>Alternative method: According to work-energy theorem, $\mathrm{W}=\Delta \mathrm{K}=0$
<br/> <br/>(Since initial and final speeds are zero)
<br/> <br/>$\therefore$ Work done by friction $+$ Work done by gravity $=0$ i.e., $-(\mu \mathrm{mg} \cos \phi) \frac{\ell}{2}+\mathrm{mg} \ell \sin \phi=0$
<br/> <br/>or $\frac{\mu}{2} \cos \phi=\sin \phi$ or $\mu=2 \tan \phi$</div>


tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
the-valency-shell-of-element-a-contains-3-electrons-while-the-valency-shell-of-element-b-contains-6-electrons-if-a-combines-with-b-the-probable-formul

slug:
the-valency-shell-of-element-a-contains-3-electrons-while-the-valency-shell-of-element-b-contains-6-electrons-if-a-combines-with-b-the-probable-formul-41490

question:
<div class="question">The valency shell of element A contains 3 electrons while the valency shell of element $\mathrm{B}$ contains 6 electrons. If A combines with $\mathrm{B}$, the probable formula of the compound formed will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{AB}_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$A_{2} B$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{A}_{2} \mathrm{~B}_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{A}_{3} \mathrm{~B}_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{A}_{2} \mathrm{~B}_{3}$</span> </div>

answer:
<div class="solution">The element A is $\mathrm{ns}^{2} p^{1}$ and $\mathrm{B}$ is $\mathrm{ns}^{2} p^{4}$. They can form compound of the type $\mathrm{A}_{2} \mathrm{~B}_{3}$</div>


title: the-value-of-0-1-cot-1-1-x-x-2-d-x-is
slug: the-value-of-0-1-cot-1-1-x-x-2-d-x-is-60689
tags: ['Mathematics', 'Definite Integration', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>x</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>-</mo><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>+</mo><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>-</mo><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mi>x</mi><mo>+</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>+</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mi>d</mi><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mi>d</mi><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mi>d</mi><mi>x</mi></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mi>a</mi></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>a</mi></msubsup><mi>f</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>d</mi><mi>x</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><mn>2</mn><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mi>d</mi><mi>x</mi></math></p><p>On evaluating by integration by parts, we have</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><mn>2</mn><mfenced close="}" open="{"><mrow><msubsup><mfenced close="]" open="["><mrow><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>·</mo><mi>x</mi></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup><mo>-</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mi>x</mi><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mi>d</mi><mi>x</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><mn>2</mn><mfenced close="}" open="{"><mrow><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>-</mo><msubsup><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><mn>2</mn><mfenced close="]" open="["><mrow><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></mrow></mfenced><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>-</mo><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mi>cot</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>-</mo><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></p><p>So, option <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>b</mi><mo>)</mo></math> is correct.</p></div>


title: the-value-of-2-1-2-4-2-2-4-6-3-is
slug: the-value-of-2-1-2-4-2-2-4-6-3-is-25122
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mn>1</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mo>+</mo><mn>4</mn></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>6</mn></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>…</mo><mo>.</mo></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>e</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>e</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>e</mi></math></span> </div>

answer:
<div class="solution"><p>Here,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>n</mi><mo>-</mo><mn>1</mn><mo>+</mo><mn>2</mn></mrow><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mn>2</mn><mo>)</mo><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>!</mo></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>S</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msub><mi>T</mi><mi>n</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><mfrac><mn>1</mn><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mn>2</mn><mo>)</mo><mo>!</mo></mrow></mfrac><mo>+</mo><mn>2</mn><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><mfrac><mn>1</mn><mrow><mo>(</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>!</mo></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>e</mi><mo>+</mo><mn>2</mn><mi>e</mi><mo>=</mo><mn>3</mn><mi>e</mi></math></p></div>


title: the-value-of-4-3-16-15-64-63-upto-n-terms-is
slug: the-value-of-4-3-16-15-64-63-upto-n-terms-is-22363
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of $\frac{3}{4}+\frac{15}{16}+\frac{63}{64}+\ldots$ upto n terms is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$n-\frac{4^{n}}{3}-\frac{1}{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{n}+\frac{4^{-\mathrm{n}}}{3}-\frac{1}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$n+\frac{4^{n}}{3}-\frac{1}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{n}-\frac{4^{-\mathrm{n}}}{3}+\frac{1}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{n}+\frac{4^{-\mathrm{n}}}{3}-\frac{1}{3}$</span> </div>

answer:
<div class="solution">Consider $\frac{3}{4}+\frac{15}{16}+\frac{63}{64}+\ldots$ upto $\mathrm{n}$ terms
<br/> <br/>$=\frac{2^{2}-1}{2^{2}}+\frac{2^{4}-1}{2^{4}}+\frac{2^{6}-1}{2^{6}}$ upto n term
<br/> <br/>$=\left(1-\frac{1}{2^{2}}\right)+\left(1-\frac{1}{2^{4}}\right)+\left(1-\frac{1}{2^{6}}\right)+\ldots$ upton
<br/> <br/>terms
<br/> <br/>$=(1+1+1+\ldots .$ upto $\mathrm{n}$ terms $)$
<br/> <br/>$-\left(\frac{1}{2^{2}}+\frac{1}{2^{4}}+\frac{1}{2^{6}}+\ldots\right.$ upto n terms $)$
<br/> <br/>$=\mathrm{n}-\frac{1}{2^{2}}\left[\frac{1-\left(\frac{1}{2^{2}}\right)^{\mathrm{n}}}{1 \frac{1}{2^{2}}}\right]=\mathrm{n}-\frac{1}{3}\left(1-4^{-\mathrm{n}}\right)$
<br/> <br/>$=\mathrm{n}+\frac{4^{-\mathrm{n}}}{3}-\frac{1}{3}$</div>


title: the-value-of-a-in-order-that-fxsin-xcos-xa-xb-decreases-for-all-real-values-is-given-by
slug: the-value-of-a-in-order-that-fxsin-xcos-xa-xb-decreases-for-all-real-values-is-given-by-30001
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of \(a\) in order that \(f(x)=\sin x-\cos x-a x+b\) decreases for all real values is given by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(a \geq \sqrt{2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a \(  &lt; \sqrt{2}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(a \geq 1\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(a  &lt; 1\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(a \geq \sqrt{2}\)</span> </div>

answer:
<div class="solution">We have; \(f(x)=\sin x-\cos x-a x+b\)<br/>\(\Rightarrow f^{\prime}(x)=\cos x+\sin x-a\)<br/>\(\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})  &lt; 0 ~\forall~ \mathrm{x} \in \mathrm{R}\)<br/>\(\Rightarrow(\cos x+\sin x) &lt; a ~\forall~ x \in R\)<br/>As the max. value of \((\cos x+\sin x)\) is \(\sqrt{2}\)<br/>The above is possible when \(a \geq \sqrt{2}\)</div>


title: the-value-of-a-so-that-lim-x-0-1-x-2-e-a-x-e-x-x-3-2-is
slug: the-value-of-a-so-that-lim-x-0-1-x-2-e-a-x-e-x-x-3-2-is-78085
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mfenced><mrow><msup><mi>e</mi><mrow><mi>a</mi><mi>x</mi></mrow></msup><mo>-</mo><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math>, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li> </ul>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mfenced><mrow><msup><mi>e</mi><mrow><mi>a</mi><mi>x</mi></mrow></msup><mo>-</mo><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>a</mi><msup><mi>e</mi><mrow><mi>a</mi><mi>x</mi></mrow></msup><mo>-</mo><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></mfenced><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> (using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>'Hospital rule to the left hand side)</p><p>As <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mn>0</mn></math>, the denominator is zero and as the value of the limit is finite so, numerator must be zero for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mn>0</mn></math>.</p><p>Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>-</mo><mn>1</mn><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>a</mi><mo>=</mo><mn>2</mn></math>.</p></div>


title: the-value-of-cos-1-xcos-1leftfracx2frac12-sqrt33-x2right-frac12-leq-x-leq-1-is
slug: the-value-of-cos-1-xcos-1leftfracx2frac12-sqrt33-x2right-frac12-leq-x-leq-1-is-20211
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of \(\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}\right)\); \(\frac{1}{2} \leq x \leq 1\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(-\frac{\pi}{3}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{\pi}{3}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{3}{\pi}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(-\frac{3}{\pi}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\pi}{3}\)</span> </div>

answer:
<div class="solution">Let \(\cos ^{-1} x=y\)<br/>\(\Rightarrow x=\cos y\), so that \(\frac{1}{2} \leq x \leq 1\) or \(\quad 0 \leq y \leq \frac{\pi}{3}\)<br/>and \(\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}=\frac{1}{2} \cos y+\frac{\sqrt{3}}{2} \sin y\)<br/>\(=\cos \frac{\pi}{3} \cos y+\sin \frac{\pi}{3} \sin y=\cos \left(\frac{\pi}{3}-y\right)\)<br/>\(\Rightarrow \cos ^{-1}\left(\frac{x}{2}+\frac{1}{2} \sqrt{3-3 x^2}\right)=\frac{\pi}{3}-y\)<br/>\(\therefore\) the given expression is equal to<br/>\(y+\frac{\pi}{3}-y, \text { i.e., } \frac{\pi}{3}\)</div>


title: the-value-of-cos-2-1-cos-1-cos-sin-1-8-63-is
slug: the-value-of-cos-2-1-cos-1-cos-sin-1-8-63-is-20800
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of
<br/> <br/>$\cos \left[\frac{1}{2} \cos ^{-1}\left(\cos \left(\sin ^{-1} \frac{\sqrt{63}}{8}\right)\right)\right]$ is $-$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 / 16$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 / 8$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3 / 4$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 / 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 / 4$</span> </div>

answer:
<div class="solution">The given trigonometric ratio
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=\cos \left[\frac{1}{2}\left(\cos \left(\cos ^{-1} \frac{1}{8}\right)\right)\right] \\
<br/> <br/>=\cos \left(\frac{1}{2} \cos ^{-1} \frac{1}{8}\right) \\
<br/> <br/>=\sqrt{\frac{1+\cos \left(\cos ^{-1} \frac{1}{8}\right)}{2}}=\frac{3}{4}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: the-value-of-cos-2-cos-1-x-sin-1-x-at-x-5-1-is-1
slug: the-value-of-cos-2-cos-1-x-sin-1-x-at-x-5-1-is-1-77595
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\frac{2 \sqrt{6}}{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-2 \sqrt{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{\sqrt{6}}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{2 \sqrt{6}}{5}$</span> </div>

answer:
<div class="solution"><br/>$\begin{aligned} &amp; \cos \left[2 \cos ^{-1} x+\sin ^{-1} x\right] \\ &amp; =\cos \left[\cos ^{-1} x+\cos ^{-1} x+\sin ^{-1} x\right] \\ &amp; =\cos \left[\cos ^{-1} x+\pi / 2\right]=-\sin \left[\cos ^{-1} x\right] \\ &amp; =-\sin \left[\sin ^{-1} \sqrt{1-x^2}\right]=-\sqrt{1-x^2} \\ &amp; =-\sqrt{1-\left(\frac{1}{5}\right)^2}=-\sqrt{\frac{24}{25}}=-\frac{2 \sqrt{6}}{5} \\ &amp; \end{aligned}$</div>


title: the-value-of-cos-2-cos-1-x-sin-1-x-at-x-5-1-is
slug: the-value-of-cos-2-cos-1-x-sin-1-x-at-x-5-1-is-73814
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of $\cos \left(2 \cos ^{-1} x+\sin ^{-1} x\right)$ at $x=\frac{1}{5}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\frac{2 \sqrt{6}}{5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-2 \sqrt{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{\sqrt{6}}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{2 \sqrt{6}}{5}$</span> </div>

answer:
<div class="solution">$\cos \left[2 \cos ^{-1} x+\sin ^{-1} x\right]$
<br/> <br/>$=\cos \left[\cos ^{-1} x+\cos ^{-1} x+\sin ^{-1} x\right]$
<br/> <br/>$=\cos \left[\cos ^{-1} x+\frac{\pi}{2}\right]=-\sin \left[\cos ^{-1} x\right]$
<br/> <br/>$=-\sin \left[\sin ^{-1} \sqrt{1-x^{2}}\right]=-\sqrt{1-x^{2}}$
<br/> <br/>$=-\sqrt{1-\left(\frac{1}{5}\right)^{2}}=-\sqrt{\frac{24}{25}}=-\frac{2 \sqrt{6}}{5}$</div>


title: the-value-of-cos-lo-g-x-dx-is
slug: the-value-of-cos-lo-g-x-dx-is-19172
question: <div class="question">The value of $\int \cos (\log x) \mathrm{d} \mathrm{x}$ is :</div>
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}[\sin (\log x)+\cos (\log x)]+C$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{x}{2}[\sin (\log x)+\cos (\log x)]+C$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{x}{2}[\sin (\log x)-\cos (\log x)]+C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{2}[\sin (\log x)-\cos (\log x)]+C$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x}{2}[\sin (\log x)+\cos (\log x)]+C$</span> </div>

answer:
<div class="solution">Let
<br/> <br/>\(\mathrm{I}=\int \cos (\log \mathrm{x}) \cdot 1 \mathrm{~d} \mathrm{x} \ldots . . \text { (i) }\)
<br/> <br/>Use integral by parts,
<br/> <br/>\(\begin{aligned} &amp;I=\cos (\log x) \cdot x-\int[-\sin (\log x)] \cdot \frac{1}{x} \cdot x d x \\ &amp;=x \cdot \cos (\log x)+\int \sin (\log x) \cdot 1 d x \\ &amp;=x \cdot \cos (\log x)+\left[\sin (\log x) \cdot x-\int \cos (\log x) \cdot \frac{1}{x} \cdot x d x\right]+C \\ &amp;=x \cdot(\log x)+\left[x \sin (\log x)-\int \cos (\log x) d x\right]+C \\ &amp;=x\{\sin (\log x)+\cos (\log x)\}-I+C[f r o m \text { Eq. (i) }] \\ &amp;\Rightarrow I=\frac{x}{2}\{\sin (\log x)+\cos (\log x)\}+C \end{aligned}\)</div>


title: the-value-of-current-i-as-shown-in-the-given-circuit-diagram-is-1
slug: the-value-of-current-i-as-shown-in-the-given-circuit-diagram-is-1-92545
tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P3.db
question:
<div class="question"><p>The value of current <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math> as shown in the given circuit diagram is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/0f2e5a9c-67bd-4935-be12-e11e8ae36e87-image.png" style="width: 200px; height: 161px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>A</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>9</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>9</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution"><p>The given circuit can be redrawn as</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/c4510ba6-fbf7-4b13-a751-dafb6ac2ade9-image.png" style="width: 180px; height: 167px;"/></p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>R</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><msub><mi>R</mi><mrow><mi>B</mi><mi>C</mi></mrow></msub></mfrac><mo>=</mo><mfrac><msub><mi>R</mi><mrow><mi>A</mi><mi>D</mi></mrow></msub><msub><mi>R</mi><mrow><mi>D</mi><mi>C</mi></mrow></msub></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo></math> hence, it is a balanced Wheatstone bridge, therefore current flowing in arm <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>D</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mi>R</mi><mi>eq</mi></msub><mo>=</mo><mfenced><mrow><mn>10</mn><mo>+</mo><mn>20</mn></mrow></mfenced><mo>∥</mo><mfenced><mrow><mn>6</mn><mo>+</mo><mn>12</mn></mrow></mfenced><mo>=</mo><mn>30</mn><mo>∥</mo><mn>18</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>30</mn><mo>×</mo><mn>18</mn></mrow><mrow><mn>30</mn><mo>+</mo><mn>18</mn></mrow></mfrac><mo>=</mo><mn>11</mn><mo>.</mo><mn>25</mn><mo> </mo><mi mathvariant="normal">Ω</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo> </mo><mo> </mo><mi>I</mi><mo>=</mo><mfrac><mi>V</mi><msub><mi>R</mi><mi>eq</mi></msub></mfrac><mo>=</mo><mfrac><mn>10</mn><mrow><mn>11</mn><mo>.</mo><mn>25</mn></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>89</mn><mo> </mo><mi>A</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>9</mn><mo> </mo><mi>A</mi></math></p></div>


tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
title:
the-value-of-f-0-so-that-the-function-f-x-2-x-sin-1-x-2-x-tan-1-x-is-continuous-at-each-point-in-its-domain-is

slug:
the-value-of-f-0-so-that-the-function-f-x-2-x-sin-1-x-2-x-tan-1-x-is-continuous-at-each-point-in-its-domain-is-15193

question:
<div class="question">The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>0</mn><mo>)</mo></math>, so that the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>-</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow></mfrac></math> is continuous at each point in its domain, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi><mo>,</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi></math> are continuous functions, so the<br/>function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is clearly continuous at each point of its domain except possibly at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.</p><p>So, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> to be continuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mn>2</mn><mo>-</mo><mfrac><mrow><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow><mi>x</mi></mfrac></mrow><mrow><mn>2</mn><mo>+</mo><mfrac><mrow><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow><mi>x</mi></mfrac></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow><mi>x</mi></mfrac><mo>=</mo><mn>1</mn><mo>,</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi>x</mi></mrow><mi>x</mi></mfrac><mo>=</mo><mn>1</mn></mrow></mfenced></math></p></div>


title: the-value-of-for-which-the-lines-3-x-4-y-13-8-x-11-y-33-and-2-x-3-y-0-are-concurrent-is
slug: the-value-of-for-which-the-lines-3-x-4-y-13-8-x-11-y-33-and-2-x-3-y-0-are-concurrent-is-23398
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-7</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of $\lambda$, for which the lines $3 x-4 y=13,8 x-11 y=33$ and $2 x-3 y+\lambda=0$
<br/> <br/>are concurrent is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">-7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{7}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">9</span> </li> </ul>

answer:
<div class="solution">For concurrency of 3 lines the determinant of coefficients of equations should be 0 .
<br/> <br/>i.e. $\left|\begin{array}{ccc}3 &amp; -4 &amp; -13 \\ 8 &amp; -11 &amp; -33 \\ 2 &amp; -3 &amp; \lambda\end{array}\right|=0$
<br/> <br/>$\Rightarrow 3(-11 \lambda-99)+4(8 \lambda+66)-13(-24+22)=0$
<br/> <br/>$\Rightarrow-33 \lambda-297+32 \lambda+264+312-286=0$
<br/> <br/>$\Rightarrow-\lambda-583+576=0 \Rightarrow \lambda=-7$</div>


title: the-value-of-for-which-the-loci-arg-z-6-and-z-2-3-i-on-the-argand-plane-touch-each-other-is
slug: the-value-of-for-which-the-loci-arg-z-6-and-z-2-3-i-on-the-argand-plane-touch-each-other-is-86297
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> for which the loci <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arg</mi><mi>z</mi><mo>=</mo><mfrac><mi>π</mi><mn>6</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>z</mi><mo>-</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mi>i</mi><mo>|</mo><mo>=</mo><mi>λ</mi></math> on the argand plane touch each other is </div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arg</mi><mi>z</mi><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>y</mi><mi>x</mi></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arg</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mi>π</mi><mn>6</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mi>y</mi><mi>x</mi></mfrac><mo>=</mo><mfrac><mi>π</mi><mn>6</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>y</mi><mi>x</mi></mfrac><mo>=</mo><mi>tan</mi><mfrac><mi>π</mi><mn>6</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>y</mi></math></p><p>Also given, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>z</mi><mo>-</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mi>i</mi><mo>|</mo><mo>=</mo><mi>λ</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mo>-</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mi>i</mi><mo>|</mo><mo>=</mo><mi>λ</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>λ</mi><mn>2</mn></msup></math></p><p>This is the equation of circle whose centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><msqrt><mn>3</mn></msqrt><mi>y</mi><mo>=</mo><mn>0</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>λ</mi></math>. Now, the line touches the circle</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>λ</mi><mo>=</mo><mfenced close="|" open="|"><mfrac><mrow><mn>0</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mfenced><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfenced></mrow><msqrt><mn>1</mn><mo>+</mo><mn>3</mn></msqrt></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mfenced close="|" open="|"><mfrac><mn>6</mn><mn>2</mn></mfrac></mfenced><mo>=</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>λ</mi><mo>=</mo><mn>3</mn></math></p></div>


title: the-value-of-leftbeginarrayccc1-2-3-4-3-6-2-7-9endarrayright-is
slug: the-value-of-leftbeginarrayccc1-2-3-4-3-6-2-7-9endarrayright-is-27663
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">231</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of \(\left|\begin{array}{ccc}1 &amp; 2 &amp; 3 \\ -4 &amp; 3 &amp; 6 \\ 2 &amp; -7 &amp; 9\end{array}\right|\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">213</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">-231</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">231</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">39</span> </li> </ul>

answer:
<div class="solution">\(\begin{aligned} &amp; \left|\begin{array}{ccc}1 &amp; 2 &amp; 3 \\ -4 &amp; 3 &amp; 6 \\ 2 &amp; -7 &amp; 9\end{array}\right|=1\left|\begin{array}{cc}3 &amp; 6 \\ -7 &amp; 9\end{array}\right|-2\left|\begin{array}{cc}-4 &amp; 6 \\ 2 &amp; 9\end{array}\right|+3 \\ &amp; \left|\begin{array}{cc}-4 &amp; 3 \\ 2 &amp; -7\end{array}\right| \\ &amp; \end{aligned}\)<br/>\(\begin{aligned}<br/>&amp; =1(3 \times 9-6(-7))-2(-4 \times 9-2 \times 6)+3 \\<br/>&amp; ((-4)(-7)-3 \times 2) \\<br/>&amp; =(27+42)-2(-36-12)+3(28-6)=231<br/>\end{aligned}\)</div>


title: the-value-of-lim-x-0-x-2-1-x-x-1-e-2-1-e-x-is
slug: the-value-of-lim-x-0-x-2-1-x-x-1-e-2-1-e-x-is-61689
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of $\lim _{x \rightarrow 0} \frac{(1+x)^{\frac{1}{x}}-e+\frac{1}{2} e x}{x^2}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$\frac{11}{24} e$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$-\frac{11}{24} e$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{e}{24}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\frac{11}{24} e$
<br/></span> </div>

answer:
<div class="solution">Let $y=(1+x)^{1 / x}$
<br/>$\Rightarrow \log y=\frac{1}{x} \log (1+x)=\frac{1}{x}\left[x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots\right]$
<br/>$=1-\frac{x}{2}+\frac{x^2}{3}-\frac{x^3}{4}+\ldots$
<br/>$\begin{aligned} y &amp; =e^{\left(1-\frac{x}{2}+\frac{x^2}{3}+\ldots .\right)}=e e^{\left(\frac{-x}{2}+\frac{x^2}{3} \ldots . .\right)} \\ &amp; =e\left[1+\left(-\frac{x}{2}+\frac{x^2}{3}-\ldots\right)+\frac{1}{2 !}\left(-\frac{x}{2}+\frac{x^2}{3}-\ldots\right)^2+\ldots\right]\end{aligned}$
<br/>$y-e+\frac{1}{2} e x=e x^2\left[\frac{1}{3}+0(x)+\frac{1}{2}\left(-\frac{1}{2}+0(x)\right)^2+\ldots.\right]$
<br/>$[\because 0(x)$ in terms containing $x]$
<br/>$\lim \frac{y-e+\frac{1}{2} e x}{x^2}=e\left[\frac{1}{3}+\frac{1}{8}\right]=\frac{11}{24} e$</div>


title: the-value-of-lim-x-0-x-4-1-c-o-s-1-c-o-s-x-is
slug: the-value-of-lim-x-0-x-4-1-c-o-s-1-c-o-s-x-is-82903
question: <div class="question">The value of $\lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^4}$ is</div>
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\frac{1}{6}$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{1}{8}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{1}{10}$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$\frac{1}{12}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\frac{1}{8}$
<br/></span> </div>

answer:
<div class="solution">$1-\cos (1-\cos x)=2 \sin ^2\left(\frac{1-\cos x}{2}\right)$
<br/>$=2 \sin ^2\left(\sin ^2 \frac{x}{2}\right)$
<br/>$\therefore \lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^4}=\lim _{x \rightarrow 0} \frac{2 \sin ^2\left(\sin ^2 \frac{x}{2}\right)}{x^4}$
<br/>$=\lim _{x \rightarrow 0} \frac{2 \sin ^2\left(\sin ^2 \frac{x}{2}\right)}{\left(\sin ^2 \frac{x}{2}\right)^2} \times \frac{\sin ^4 \frac{x}{2}}{\left(\frac{x}{2}\right)^4 \times 16}$
<br/>$=\frac{1}{8}$</div>


title: the-value-of-lim-x-3-4-4-sin-2-x-cos-x-cos-x-sin-x-sin-x-cos-x-is
slug: the-value-of-lim-x-3-4-4-sin-2-x-cos-x-cos-x-sin-x-sin-x-cos-x-is-61460
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>4</mn></mfrac></mrow></munder><mfrac><mrow><mn>4</mn><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi><mi>cos</mi><mi>x</mi><mo>-</mo><mi>cos</mi><mi>x</mi><mo>+</mo><mi>sin</mi><mi>x</mi></mrow><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfrac></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>4</mn></mfrac></mrow></munder><mfrac><mrow><mn>4</mn><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi><mi>cos</mi><mi>x</mi><mo>-</mo><mi>cos</mi><mi>x</mi><mo>+</mo><mi>sin</mi><mi>x</mi></mrow><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>4</mn></mfrac></mrow></munder><mfrac><mrow><mn>2</mn><mi>sin</mi><mi>x</mi><mfenced><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi>x</mi><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mi>x</mi><mo>+</mo><mn>2</mn><mi>sin</mi><mi>x</mi><mi>cos</mi><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>cos</mi><mi>x</mi><mo>+</mo><mi>sin</mi><mi>x</mi></mrow><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>4</mn></mfrac></mrow></munder><mfrac><mrow><mn>2</mn><mi>sin</mi><mi>x</mi><mfenced close="]" open="["><mrow><mo>(</mo><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>cos</mi><mi>x</mi><mo>+</mo><mi>sin</mi><mi>x</mi></mrow><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>4</mn></mfrac></mrow></munder><mfrac><mrow><mn>2</mn><mi>sin</mi><mi>x</mi><mo>(</mo><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi><msup><mo>)</mo><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>sin</mi><mi>x</mi><mo>-</mo><mi>cos</mi><mi>x</mi><mo>+</mo><mi>sin</mi><mi>x</mi></mrow><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>4</mn></mfrac></mrow></munder><mfrac><mrow><mfenced><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfenced><mfenced close="}" open="{"><mrow><mn>2</mn><mi>sin</mi><mi>x</mi><mfenced><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>4</mn></mfrac></mrow></munder><mn>2</mn><mi>sin</mi><mi>x</mi><mfenced><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfenced><mo>-</mo><mn>1</mn><mo>=</mo><mo>-</mo><mn>1</mn></math></p></div>


title: the-value-of-lim-x-rightarrow-0-fracleft4x1right3sin-fracx24-log-13-x-is
slug: the-value-of-lim-x-rightarrow-0-fracleft4x1right3sin-fracx24-log-13-x-is-25656
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of \(\lim _{x \rightarrow 0} \frac{\left(4^x-1\right)^3}{\sin \frac{x^2}{4} \log (1+3 x)}\), is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{4}{3}(\ln 4)^2\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{4}{3}(\ln 4)^3\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{3}{2}(\ln 4)^2\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{3}{2}(\ln 4)^3\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{4}{3}(\ln 4)^3\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \lim _{x \rightarrow 0} \frac{\left(4^x-1\right)^3}{\sin \frac{x^2}{4} \log (1+3 x)} \\<br/>&amp; =\lim _{x \rightarrow 0} \frac{\left(4^x-1\right)^3}{x^3} \cdot \frac{(x / 2)^2}{\sin x^2 / 4} \cdot \frac{3 x}{\log (1+3 x)} \cdot \frac{4}{3} \\<br/>&amp; =\frac{4}{3}\left(\log _e 4\right)^3 \cdot 1 \cdot \log _e(e)=\frac{4}{3}\left(\log _e 4\right)^3 .<br/>\end{aligned}\)</div>


title: the-value-of-r-1-10-r-c-r-n-c-r-1-n-is-equal-to
slug: the-value-of-r-1-10-r-c-r-n-c-r-1-n-is-equal-to-41765
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mi>r</mi><mo>·</mo><mfrac><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><mmultiscripts><mi>C</mi><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts></mfrac></math> is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>9</mn></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mi>n</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>(</mo><mi>n</mi><mo>-</mo><mn>4</mn><mo>)</mo></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>9</mn></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mi>r</mi><mo>·</mo><mfrac><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><mmultiscripts><mi>C</mi><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts></mfrac><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mfenced><mrow><mi>n</mi><mo>-</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mi>r</mi><mo>·</mo><mfrac><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><mmultiscripts><mi>C</mi><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts></mfrac><mo>=</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mfenced close="}" open="{"><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>r</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mi>r</mi><mo>·</mo><mfrac><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><mmultiscripts><mi>C</mi><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts></mfrac><mo>=</mo><mn>10</mn><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mi>r</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><munderover><mo>∑</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>10</mn></munderover><mi>r</mi><mo>·</mo><mfrac><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><mmultiscripts><mi>C</mi><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts></mfrac><mo>=</mo><mn>10</mn><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>55</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>10</mn><mi>n</mi><mo>-</mo><mn>45</mn><mo>=</mo><mn>5</mn><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>9</mn></mrow></mfenced></math></p></div>


title: the-value-of-s-n-1-tan-1-2-n-n-4-n-2-2-is-equal-to
slug: the-value-of-s-n-1-tan-1-2-n-n-4-n-2-2-is-equal-to-11201
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><msup><mi>n</mi><mn>4</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>2</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>4</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>4</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><msup><mi>n</mi><mn>4</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>4</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><mfenced close="]" open="["><mrow><msup><mfenced><msup><mi>n</mi><mn>2</mn></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><msup><mi>n</mi><mn>2</mn></msup></mfenced><mfenced><mn>1</mn></mfenced></mrow></mfenced><mo>-</mo><msup><mi>n</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mi>n</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><msup><mi>n</mi><mn>4</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math> </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><msup><mi>n</mi><mn>4</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>iii</mi></mfenced></math></p><p>On putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>…</mo></math> successively in Eq. (iii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>3</mn><mo>-</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>7</mn><mo>-</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>3</mn></msub><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>13</mn><mo>-</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>7</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math></p><p>On adding vertically, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msub><mi>u</mi><mi>n</mi></msub></math>   [from Eq. (i)]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>=</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></math></p></div>


title: the-value-of-sin-left4-tan-1-frac13rightcos-left2-tan-1-frac17right-is
slug: the-value-of-sin-left4-tan-1-frac13rightcos-left2-tan-1-frac17right-is-66094
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of \(\sin \left(4 \tan ^{-1} \frac{1}{3}\right)-\cos \left(2 \tan ^{-1} \frac{1}{7}\right)\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(3 / 7\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(7 / 8\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(8 / 21\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">\(\sin \left(4 \tan ^{-1} \frac{1}{3}\right)\)<br/>\(\begin{aligned}<br/>&amp;= 2 \sin \left(2 \tan ^{-1} \frac{1}{3}\right) \cos \left(2 \tan ^{-1} \frac{1}{3}\right) \\<br/>&amp;=2 \sin \left(\tan ^{-1} \frac{3}{4}\right)=\cos \left(\tan ^{-1} \frac{3}{4}\right)=2 \cdot \frac{3}{5} \cdot \frac{4}{5}=\frac{24}{25} \\<br/>&amp; \cos \left(2 \tan ^{-1} \frac{1}{7}\right)=\cos \left(\tan ^{-1} \frac{7}{24}\right)=\frac{24}{25} .<br/>\end{aligned}\)<br/>The given expression \(=0\)</div>


title: the-value-of-sin-n-1-n-4-n-i-is
slug: the-value-of-sin-n-1-n-4-n-i-is-62370
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of $\sin \left[\mathrm{n} \pi+(-1)^{\mathrm{n}} \frac{\pi}{4}\right], \mathrm{n} \in \mathrm{I}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\sqrt{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{1}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\sqrt{2}}$</span> </div>

answer:
<div class="solution">$\sin \left[\mathrm{n} \pi+(-1)^{\mathrm{n}} \frac{\pi}{4}\right]=(-1)^{\mathrm{n}} \sin \left[(-1)^{\mathrm{n}} \frac{\pi}{4}\right]$
<br/> <br/>$\left[\because \sin (\mathrm{n} \pi+\theta)=(-1)^{\mathrm{n}} \sin \theta\right]$
<br/> <br/>$\begin{aligned} &amp;=(-1)^{\mathrm{n}}(-1)^{\mathrm{n}} \sin \frac{\pi}{4} \\ \therefore \sin &amp;\left.\left[(-1)^{\mathrm{n}} \theta\right]=(-1)^{\mathrm{n}} \sin \theta\right] \\ &amp;=(-1)^{2 \mathrm{n}} \sin \frac{\pi}{4}=\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}} \end{aligned}$</div>


title: the-value-of-the-determinant-265-240-219-240-225-198-219-198-181-is
slug: the-value-of-the-determinant-265-240-219-240-225-198-219-198-181-is-87472
tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of the determinant 
<br/>$\left|\begin{array}{lll}265 &amp; 240 &amp; 219 \\ 240 &amp; 225 &amp; 198 \\ 219 &amp; 198 &amp; 181\end{array}\right|$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1000</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">779</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">679</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Applying $\mathrm{C}_{1}-\mathrm{C}_{2}$ and $\mathrm{C}_{2}-\mathrm{C}_{3},$ we get
<br/>$\begin{aligned}
<br/>\text { Det. } &amp;=\left|\begin{array}{ccc}
<br/>25 &amp; 21 &amp; 219 \\
<br/>15 &amp; 27 &amp; 198 \\
<br/>21 &amp; 17 &amp; 181
<br/>\end{array}\right|=\left|\begin{array}{ccc}
<br/>4 &amp; 21 &amp; 9 \\
<br/>-12 &amp; 27 &amp; -72 \\
<br/>4 &amp; 17 &amp; 11
<br/>\end{array}\right| \\
<br/>=&amp;\left[\begin{array}{llc}
<br/>4 &amp; 21 &amp; 9 \\
<br/>0 &amp; 90 &amp; -45 \\
<br/>0 &amp; -4 &amp; 2
<br/>\end{array} \mid\left[\mathrm{By} \mathrm{R}_{2}+3 \mathrm{R}_{1}, \mathrm{R}_{3}-\mathrm{R}_{1}\right]\right.\\
<br/>=4(180-180)=0
<br/>\end{aligned}$</div>


title: the-value-of-x-1-3-x-2-5-4-1-1-d-x-is
slug: the-value-of-x-1-3-x-2-5-4-1-1-d-x-is-23855
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
question:
<div class="question">The value of $\int \frac{1}{\left[(x-1)^3(x+2)^5\right]^{\frac{1}{4}}} d x$, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$\frac{4}{3}\left(\frac{x+1}{x-2}\right)^{\frac{1}{4}}+C$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$\frac{3}{4}\left(\frac{x-1}{x+2}\right)^{\frac{1}{4}}+C$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{\frac{1}{4}}+C$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$\frac{1}{3}\left(\frac{2 x-1}{4 x-3}\right)^{\frac{1}{4}}+C$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{\frac{1}{4}}+C$
<br/></span> </div>

answer:
<div class="solution">$I=\int \frac{1}{\left(\frac{x-1}{x+2}\right)^{3 / 4}(x+2)^2} d x$
<br/>Let $\frac{x-1}{x+2}=t \Rightarrow \frac{3 d x}{(x+2)^2}=d t$
<br/>$$
<br/>\begin{aligned}
<br/>\Rightarrow \quad I &amp; =\frac{1}{3} \int \frac{1}{t^{3 / 4}} d t=\frac{1}{3}\left(\frac{t^{1 / 4}}{1 / 4}\right)+C \\
<br/>&amp; =\frac{4}{3} t^{1 / 4}+C=\frac{4}{3}\left(\frac{x-1}{x+2}\right)^{1 / 4}+C
<br/>\end{aligned}
<br/>$$</div>


title: the-value-of-x-is-maximum-for
slug: the-value-of-x-is-maximum-for-48315
question: <div class="question">The value of $\mathrm{x}$ is maximum for</div>
tags: ['Chemistry', 's Block Elements', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{MgSO}_{4} \cdot \mathrm{x} \mathrm{H}_{2} \mathrm{O}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{CaSO}_{4} \cdot \mathrm{x} \mathrm{H}_{2} \mathrm{O}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{BaSO}_{4} \cdot \mathrm{x} \mathrm{H}_{2} \mathrm{O}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">All have the same</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{MgSO}_{4} \cdot \mathrm{x} \mathrm{H}_{2} \mathrm{O}$</span> </div>

answer:
<div class="solution">Because of smaller size, $\mathrm{Mg}^{2+}$ ions are extensively hydrated.</div>


tags: ['Chemistry', 'Solutions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
title:
the-vant-hoff-factor-i-for-an-electrolyte-which-undergoes-dissociation-and-association-in-solvents-are-respectively

slug:
the-vant-hoff-factor-i-for-an-electrolyte-which-undergoes-dissociation-and-association-in-solvents-are-respectively-23977

question:
<div class="question">The van't Hoff factor i for an electrolyte which undergoes dissociation and association in solvents are respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">greater than 1 and greater than 1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">less than 1 and greater than 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">less than 1 and less than 1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">greater than 1 and less than 1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">greater than 1 and less than 1</span> </div>

answer:
<div class="solution">When an electrolyte dissociates van't Hoff factor i is greater than 1 and when it associates the $\mathrm{i}$ is less than $1.$</div>


title: the-vapour-density-of-ozone-is
slug: the-vapour-density-of-ozone-is-97720
question: <div class="question">The vapour density of ozone is</div>
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">24</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">16</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">32</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">24</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">48</span> </li> </ul>

answer:
<div class="solution">We know that, Molecular weight of compound or molecules $=2 \times \mathrm{V}$. $\mathrm{D}$<br/>Vapour density (V. D.) of ozone molecules<br/>$=\frac{\mathrm{M} . \text { wt. of } \mathrm{O}_{3}}{2}=\frac{48}{2}=24$<br/>Hence, $\mathrm{V} . \mathrm{D}$. of $\mathrm{O}_{3}$ is 24</div>


title: the-vapour-pressure-of-is-higher-than-due-to
slug: the-vapour-pressure-of-is-higher-than-due-to-32819
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
question:
<div class="question">The vapour pressure of
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/-f7gWA2egEV8jcvJ5Uvfr_P5oMbYmwgeVhguXPezdZc.original.fullsize.png"/><br/> <br/>is higher than
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9RPwDWv2fLFqbgvWByU00cU9GDISFpdEXthNc2aJcAY.original.fullsize.png"/><br/> <br/>due to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Dipole moment</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Dipole- dipole interaction</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}$ - bonding</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Lattice structure</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{H}$ - bonding</span> </div>

answer:
<div class="solution">Ortho-nitrophenol has intramolecular
<br/>H bonding 
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Rt43sBNInI87vhCPxQ2h5ieDrSuBJOUrJeIszsEbVIM.original.fullsize.png"/><br/> <br/>and paranitrophenol has intermolecular H-bonding.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/W6V28xNUc5zQOAq06FtYvBrtaB9UiMs9iMRDWELnDdc.original.fullsize.png"/><br/> <br/>Hence former is more volatile than latter.</div>


tags: ['Chemistry', 'Solutions', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.1</span> </div>
source_db: MarksBatch2_P3.db
title:
the-vapour-pressure-of-two-pure-liquids-a-and-b-that-form-an-ideal-solution-are-400-and-800-mm-of-hg-respectivelyat-a-temperature-t-c-the-mole-fractio

slug:
the-vapour-pressure-of-two-pure-liquids-a-and-b-that-form-an-ideal-solution-are-400-and-800-mm-of-hg-respectivelyat-a-temperature-t-c-the-mole-fractio-17823

question:
<div class="question">The vapour pressure of two pure liquids $A$ and $B$ that form an ideal solution, are 400 and $800 \mathrm{~mm}$ of Hg respectivelyat a temperature $t^{\circ} \mathrm{C}$. The mole fraction of $A$ in a solution of $A$ and $B$ whose boiling point is $t^{\circ} \mathrm{C}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.8</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0.1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.2</span> </li> </ul>

answer:
<div class="solution">V.P. of solution at $t^{\circ} \mathrm{C}=760 \mathrm{~mm}$ [at b.p., V.P. of solution =atompheric pressure] Thus $=\mathrm{P}_{\mathrm{A}}^{\circ} \mathrm{x}_{\mathrm{A}}+\mathrm{P}_{\mathrm{B}}^{\infty} \mathrm{x}_{\mathrm{B}}$
<br/> <br/>or $\mathrm{P}=\mathrm{P}_{\mathrm{A}}^{0} \cdot \mathrm{x}_{\mathrm{A}}+\mathrm{P}_{\mathrm{B}}^{0}\left(1-\mathrm{x}_{\mathrm{A}}\right)\left[\because \mathrm{x}_{\mathrm{A}}+\mathrm{x}_{\mathrm{B}}=1\right]$
<br/> <br/>or $760=400 \mathrm{X}_{\mathrm{A}}+800\left(1-\mathrm{X}_{\mathrm{A}}\right)[\because \mathrm{P}=760 \mathrm{mmofHg}]$
<br/> <br/>or $-800+760=-400 x_{\mathrm{A}}$
<br/> <br/>or $-40=-400 \mathrm{x}_{\mathrm{A}}$
<br/> <br/>or $x_{\mathrm{A}}=\frac{40}{400}=0.1$
<br/> <br/>Thus mole fraction in solution is 0.1</div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
title:
the-variation-of-kinetic-energy-ke-of-a-particle-executing-simple-harmonic-motion-with-the-displacement-x-starting-from-mean-position-to-extreme-posit-1

slug:
the-variation-of-kinetic-energy-ke-of-a-particle-executing-simple-harmonic-motion-with-the-displacement-x-starting-from-mean-position-to-extreme-posit-1-19600

question:
<div class="question">The variation of kinetic energy (KE) of a particle executing simple harmonic motion with the displacement (x) starting from mean position to extreme position (A) is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/dimh0q40udXvWSKmHpJiz8xNY5YVjqr58-F8id9cL78.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/mUHWi2SDOUu5yjq4TzfhE5FuHlTJuJxkWwYc2iq10og.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/I-X5KKseNU8RJmOtJxz_fCkQlR6SNsMC4V3bIhZyO_8.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/K7iMRNXzppX1JMP1jz93GejVSKH77LJ8VE6XDuY9M8o.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/K7iMRNXzppX1JMP1jz93GejVSKH77LJ8VE6XDuY9M8o.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><br/>$\begin{aligned} &amp; T \cdot E=K \cdot E+P . E \\ &amp; \Rightarrow K \cdot E=T \cdot E-K \cdot E \\ &amp; =\frac{1}{2} k^2-\frac{1}{2} k^2 \\ &amp; \Rightarrow K E=\frac{1}{2} m \omega^2\left(A^2-x^2\right) \quad\left[\because k=m \omega^2\right]\end{aligned}$<br/>$\mathrm{KE}$ is maximum at $\mathrm{x}=0$ \&amp; $\mathrm{KE}$ is zero when $\mathrm{x}=\mathrm{A}$ $\therefore \mathrm{K} . \mathrm{E}$ vs $\mathrm{x}$ graph is parabola.</div>


tags: ['Physics', 'Oscillations', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-variation-of-kinetic-energy-ke-of-a-particle-executing-simple-harmonic-motion-with-the-displacement-x-starting-from-mean-position-to-extreme-posit-2

slug:
the-variation-of-kinetic-energy-ke-of-a-particle-executing-simple-harmonic-motion-with-the-displacement-x-starting-from-mean-position-to-extreme-posit-2-79459

question:
<div class="question">The variation of kinetic energy (KE) of a particle executing simple harmonic motion with the displacement (x) starting from mean position to extreme position (A) is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/dimh0q40udXvWSKmHpJiz8xNY5YVjqr58-F8id9cL78.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/mUHWi2SDOUu5yjq4TzfhE5FuHlTJuJxkWwYc2iq10og.original.fullsize.png"/><br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/I-X5KKseNU8RJmOtJxz_fCkQlR6SNsMC4V3bIhZyO_8.original.fullsize.png"/><br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/K7iMRNXzppX1JMP1jz93GejVSKH77LJ8VE6XDuY9M8o.original.fullsize.png"/><br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/K7iMRNXzppX1JMP1jz93GejVSKH77LJ8VE6XDuY9M8o.original.fullsize.png"/><br/></span> </div>

answer:
<div class="solution"><br/>$\begin{aligned} &amp; T \cdot E=K \cdot E+P . E \\ &amp; \Rightarrow K \cdot E=T \cdot E-K \cdot E \\ &amp; =\frac{1}{2} k^2-\frac{1}{2} k^2 \\ &amp; \Rightarrow K E=\frac{1}{2} m \omega^2\left(A^2-x^2\right) \quad\left[\because k=m \omega^2\right]\end{aligned}$<br/>$\mathrm{KE}$ is maximum at $\mathrm{x}=0$ \&amp; $\mathrm{KE}$ is zero when $\mathrm{x}=\mathrm{A}$ $\therefore \mathrm{K} . \mathrm{E}$ vs $\mathrm{x}$ graph is parabola.</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
the-velocity-and-acceleration-vectors-of-a-particle-undergoing-circular-motion-are-v-2-i-m-s-and-a-2-i-4-j-m-s-2-respectively-at-an-instant-of-time-th

slug:
the-velocity-and-acceleration-vectors-of-a-particle-undergoing-circular-motion-are-v-2-i-m-s-and-a-2-i-4-j-m-s-2-respectively-at-an-instant-of-time-th-21394

question:
<div class="question">The velocity and acceleration vectors of a particle undergoing circular motion are $\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$ and $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}^{2}$ respectively at an instant of time. The radius of the circle is -</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1 \mathrm{~m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \mathrm{~m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 \mathrm{~m}$</span> </div>

answer:
<div class="solution">It can be observed that component of acceleration perpendicular to velocity is $\mathrm{a}=4 \mathrm{~m} / \mathrm{s}^{2}$
<br/> <br/>$$
<br/> <br/>\therefore \text { radius }=\frac{\mathrm{v}^{2}}{\mathrm{a}_{\mathrm{c}}}=\frac{(2)^{2}}{4}=1 \text { metre }
<br/> <br/>$$</div>


tags: ['Physics', 'Mathematics in Physics', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-velocity-of-a-projectile-at-the-initial-point-a-is-2-i-3-j-m-s-1-its-velocity-in-m-s-1-at-point-b-is-1

slug:
the-velocity-of-a-projectile-at-the-initial-point-a-is-2-i-3-j-m-s-1-its-velocity-in-m-s-1-at-point-b-is-1-89059

question:
<div class="question"><p>The velocity of a projectile at the initial point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></mrow></mfenced><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Its velocity (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>) at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/37ddd8a8-ec7d-4555-bea7-f806e51fc14e-image.png" style="width: 190px; height: 183px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </div>

answer:
<div class="solution"><p>From the figure, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>-component remains unchanged while the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math>-component is reverse. Then, the velocity at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></mrow></mfenced><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p></div>


tags: ['Physics', 'Mathematics in Physics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
title:
the-velocity-of-a-projectile-at-the-initial-point-a-is-2-i-3-j-m-s-1-its-velocity-in-m-s-1-at-point-b-is

slug:
the-velocity-of-a-projectile-at-the-initial-point-a-is-2-i-3-j-m-s-1-its-velocity-in-m-s-1-at-point-b-is-82678

question:
<div class="question"><p>The velocity of a projectile at the initial point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></mrow></mfenced><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>. Its velocity (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>) at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/37ddd8a8-ec7d-4555-bea7-f806e51fc14e-image.png" style="width: 190px; height: 183px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </div>

answer:
<div class="solution"><p>From the figure, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>-component remains unchanged while the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math>-component is reverse. Then, the velocity at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>3</mn><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></mrow></mfenced><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p></div>


title: the-vertices-of-the-hyperbola-9-x-2-16-y-2-36-x-96-y-252-0-are
slug: the-vertices-of-the-hyperbola-9-x-2-16-y-2-36-x-96-y-252-0-are-26245
question: <div class="question">The vertices of the hyperbola $9 x^2-16 y^2-36 x+96 y-252=0$ are</div>
tags: ['Mathematics', 'Hyperbola', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(6,3),(-2,3)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(6,3),(-6,3)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-6,3),(-6,-3)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(2,3),(-2,3)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(6,3),(-2,3)$</span> </div>

answer:
<div class="solution">Given hyperbola is<br/>$\begin{aligned}<br/>&amp; 9 x^2-16 y^2-36 x+96 y-252=0 \\<br/>&amp; \Rightarrow 9\left(x^2-4 x\right)-16\left(y^2-6 y\right)=252 \\<br/>&amp; \Rightarrow 9\left(x^2-4 x+4\right)-16\left(y^2-6 y+9\right) \\<br/>&amp; =252+36-144 \\<br/>&amp; \Rightarrow 9(x-2)^2-16(y-3)^2=144<br/>\end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/jLY8-6clWoBW7M9aqoLEdPXjxPnZv_kpDKECzHGVaUY.original.fullsize.png"/><br/><br/>Put $x=X+2$ and $y=Y+3$<br/>$\therefore$ Equation (i) becomes $\frac{X^2}{16}-\frac{Y^2}{9}=1$<br/>Now, vertices are $X= \pm a$ wehre $a=4$ and $Y=0$<br/>Hence, vertices are $(6,3),(-2,3)$.</div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
the-wave-number-of-the-limiting-line-in-lyman-series-of-hydrogen-is-109678-cm-1-the-wave-number-of-the-limiting-line-in-balmer-series-of-he-would-be

slug:
the-wave-number-of-the-limiting-line-in-lyman-series-of-hydrogen-is-109678-cm-1-the-wave-number-of-the-limiting-line-in-balmer-series-of-he-would-be-53675

question:
<div class="question">The wave number of the limiting line in Lyman series of hydrogen is $109678 \mathrm{~cm}^{-1}$. The wave number of the limiting line in Balmer series of $\mathrm{He}^{+}$ would be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$54839 \mathrm{~cm}^{-1}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$109678 \mathrm{~cm}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$219356 \mathrm{~cm}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$438712 \mathrm{~cm}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$109678 \mathrm{~cm}^{-1}$</span> </div>

answer:
<div class="solution">$\mathrm{R}_{\mathrm{H}}=109678 \mathrm{~cm}^{-1}$
<br/> <br/>Wave number of the limiting line in Balmer series of $\mathrm{He}^{+}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=\mathrm{R}_{\mathrm{H}} \cdot \mathrm{Z}^{2}\left[\frac{1}{\mathrm{n}_{1}^{2}}-\frac{1}{\mathrm{n}_{2}^{2}}\right] \\
<br/> <br/>=109678 \times(2)^{2}\left[\frac{1}{[2)^{2}}-\frac{1}{\infty}\right] \\
<br/> <br/>=109678 \mathrm{~cm}^{-1}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
title:
the-wavelength-corresponding-to-maximum-energy-for-hydrogen-is-91-2-nm-the-corresponding-wavelength-for-he-ion-is

slug:
the-wavelength-corresponding-to-maximum-energy-for-hydrogen-is-91-2-nm-the-corresponding-wavelength-for-he-ion-is-36995

question:
<div class="question">The wavelength corresponding to maximum energy for hydrogen is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>91</mn><mo>.</mo><mn>2</mn><mo> </mo><mi>nm</mi></math>. The corresponding wavelength for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>He</mi><mo>+</mo></msup></math> ion is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>28</mn><mo> </mo><mi>nm</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>22</mn><mo>.</mo><mn>8</mn><mo> </mo><mi>nm</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>182</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>nm</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>364</mn><mo>.</mo><mn>8</mn><mo> </mo><mi>nm</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>22</mn><mo>.</mo><mn>8</mn><mo> </mo><mi>nm</mi></math></span> </div>

answer:
<div class="solution"><p>Accordinq to Rydberg's equation</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">v</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mn>1</mn><mi mathvariant="normal">λ</mi></mfrac><mo>=</mo><msup><mi>RZ</mi><mn>2</mn></msup><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><msubsup><mi mathvariant="normal">n</mi><mi mathvariant="normal">L</mi><mn>2</mn></msubsup></mfrac><mo>-</mo><mfrac><mn>1</mn><msubsup><mi mathvariant="normal">n</mi><mi mathvariant="normal">H</mi><mn>2</mn></msubsup></mfrac></mrow></mfenced></math></p><p>where,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="normal">v</mi><mo>¯</mo></mover><mo>=</mo></math> wave number</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">λ</mi><mo>=</mo></math> wavelength</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mo>=</mo></math> Rydberg constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mn>109677</mn><mo> </mo><msup><mi>cm</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Z</mi><mo>=</mo></math> atomic number</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">n</mi><mi mathvariant="normal">L</mi></msub><mo>=</mo></math> lower energy state</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">n</mi><mi mathvariant="normal">H</mi></msub><mo>=</mo></math> higher energy state.</p><p>i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi mathvariant="normal">λ</mi></mfrac><mo>∝</mo><msup><mi mathvariant="normal">Z</mi><mn>2</mn></msup></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">λ</mi><mfenced><msup><mi>He</mi><mo>+</mo></msup></mfenced></mrow><mrow><mi mathvariant="normal">λ</mi><mfenced><mi mathvariant="normal">H</mi></mfenced></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">Z</mi><mn>2</mn></msup><mfenced><mi mathvariant="normal">H</mi></mfenced></mrow><mrow><msup><mi mathvariant="normal">Z</mi><mn>2</mn></msup><mfenced><msup><mi>He</mi><mo>+</mo></msup></mfenced></mrow></mfrac></math></p><p>To find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">λ</mi><mfenced><msup><mi>He</mi><mo>+</mo></msup></mfenced><mo>=</mo><mo>?</mo></math></p><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">λ</mi><mfenced><mi mathvariant="normal">H</mi></mfenced><mo>=</mo><mn>91</mn><mo>.</mo><mn>2</mn><mo> </mo><mi>nm</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Z</mi><mfenced><mi mathvariant="normal">H</mi></mfenced><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Z</mi><mfenced><mi>He</mi></mfenced><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi mathvariant="normal">λ</mi><mfenced><msup><mi>He</mi><mo>+</mo></msup></mfenced><mo>=</mo><mfrac><mrow><mn>1</mn><mo>×</mo><mn>91</mn><mo>.</mo><mn>2</mn></mrow><mn>4</mn></mfrac><mo>=</mo><mn>22</mn><mo>.</mo><mn>8</mn><mo> </mo><mi>nm</mi></math></p></div>


title: the-wavelength-of-radiation-emitted-by-a-body-depends-upon
slug: the-wavelength-of-radiation-emitted-by-a-body-depends-upon-15881
question: <div class="question">The wavelength of radiation emitted by a body depends upon</div>
tags: ['Physics', 'Thermal Properties of Matter', 'BITSAT', 'BITSAT 2014']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">the nature of its surface</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">the area of its surface</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">the temperature of its surface</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">All of the above</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">All of the above</span> </div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
the-work-done-in-blowing-a-soap-bubble-of-surface-tension-006-nm-1-from-2-cm-radius-to-5-cm-radius-is

slug:
the-work-done-in-blowing-a-soap-bubble-of-surface-tension-006-nm-1-from-2-cm-radius-to-5-cm-radius-is-12398

question:
<div class="question">The work done in blowing a soap bubble of surface tension $0.06 \times \mathrm{Nm}^{-1}$ from $2 \mathrm{~cm}$ radius to $5 \mathrm{~cm}$ radius is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.004168 \mathrm{~J}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.003168 \mathrm{~J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.003158 \mathrm{~J}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.004568 \mathrm{~J}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.003168 \mathrm{~J}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \Delta \mathrm{W} &amp;=\mathrm{S} \times \Delta \mathrm{A}=0.06 \times 4 \pi\left(\mathrm{r}_{2}^{2}-\mathrm{r}_{1}^{2}\right) \\ &amp;=0.003168 \mathrm{~J} \end{aligned}$</div>


tags: ['Physics', 'Capacitance', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
title:
the-work-done-in-placing-a-charge-of-8-1-0-18-coulomb-on-a-condenser-of-capacity-100-microfarad-is-1

slug:
the-work-done-in-placing-a-charge-of-8-1-0-18-coulomb-on-a-condenser-of-capacity-100-microfarad-is-1-11791

question:
<div class="question">The work done in placing a charge of $8 \times 10^{-18}$ coulomb on a condenser of capacity 100 microfarad is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3.1 \times 10^{-26}$ joule</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \times 10^{-10}$ joule</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$32 \times 10^{-32}$ joule</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$16 \times 10^{-32}$ joule</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$32 \times 10^{-32}$ joule</span> </div>

answer:
<div class="solution">Work done $=\frac{1}{2} \frac{\mathrm{q}^2}{\mathrm{C}}=\frac{\left(8 \times 10^{-18}\right)^2}{2 \times 100 \times 10^{-6}}$<br/>$=32 \times 10^{-32} \mathrm{~J}$</div>


title: the-youngs-modulus-of-a-perfectly-rigid-body-is
slug: the-youngs-modulus-of-a-perfectly-rigid-body-is-38050
question: <div class="question">The Young's modulus of a perfectly rigid body is</div>
tags: ['Physics', 'Mechanical Properties of Solids', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">unity</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">zero</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">infinity</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">some finite non-zero constant</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">infinity</span> </div>

answer:
<div class="solution">For a perfectly rigid body strain produced is zero for the given force applied, so $\mathrm{Y}=$ stress/strain $=\infty$</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-4,2</span> </div>
source_db: MarksBatch2_P3.db
title:
there-are-four-numbers-of-which-the-first-three-are-in-gp-and-the-last-three-are-in-ap-whose-common-difference-is-6-if-the-first-and-the-last-numbers-

slug:
there-are-four-numbers-of-which-the-first-three-are-in-gp-and-the-last-three-are-in-ap-whose-common-difference-is-6-if-the-first-and-the-last-numbers-76575

question:
<div class="question">There are four numbers of which the first three are in G.P. and the last three are in A.P., whose common difference is 6 . If the first and the last numbers are equal then two other numbers are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-2,4</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">-4,2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2,6</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">none</span> </li> </ul>

answer:
<div class="solution">Let the last three numbers in A.P. be a, a +6 , a $+12,$ then the first term is alsoa +12 . But $a+12, a, a+6$ are in G.P.
<br/> <br/>$\therefore \mathrm{a}^{2}=(\mathrm{a}+12)(\mathrm{a}+6) \Rightarrow \mathrm{a}^{2}=\mathrm{a}^{2}+18 \mathrm{a}+72$
<br/> <br/>$\therefore \mathrm{a}=-4$
<br/> <br/>$\therefore$ The numbers are 8,-4,2,8.</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4: 1</span> </div>
source_db: MarksBatch2_P3.db
title:
there-are-two-wire-of-same-material-and-same-length-while-the-diameter-of-second-wire-is-two-times-the-diameter-of-first-wire-then-the-ratio-of-extens

slug:
there-are-two-wire-of-same-material-and-same-length-while-the-diameter-of-second-wire-is-two-times-the-diameter-of-first-wire-then-the-ratio-of-extens-88725

question:
<div class="question">There are two wire of same material and same length while the diameter of second wire is two times the diameter of first wire, then the ratio of extension produced in the wires by applying same load will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1: 1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2: 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1: 2</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">4: 1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\because$ Both wires are same materials so both will have same Young's modulus, and let it be Y. $Y=\frac{\text { stress }}{\text { strain }}=\frac{F}{\text { A. }(\Delta L / L)}, F=$ applied force
<br/> <br/>$\mathrm{A}=$ area of cross-section of wire
<br/> <br/>Now,
<br/> <br/>$\mathrm{Y}_{1}=\mathrm{Y}_{2} \Rightarrow \frac{\mathrm{FL}}{\left(\mathrm{A}_{1}\right)\left(\Delta \mathrm{L}_{1}\right)}=\frac{\mathrm{FL}}{\left(\mathrm{A}_{2}\right)\left(\Delta \mathrm{L}_{2}\right)}$
<br/> <br/>Since load and length are same for both
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Rightarrow \mathrm{r}_{1}^{2} \Delta \mathrm{L}_{1}=\mathrm{r}_{2}^{2} \Delta \mathrm{L}_{2},\left(\frac{\Delta \mathrm{L}_{1}}{\Delta \mathrm{L}_{2}}\right)=\left(\frac{\mathrm{r}_{2}}{\mathrm{r}_{1}}\right)^{2}=4 \\
<br/> <br/>\Delta \mathrm{L}_{1}: \Delta \mathrm{L}_{2}=4: 1
<br/> <br/>\end{array}$</div>


tags: ['Physics', 'Gravitation', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">zero</span> </div>
answer: <div class="solution">Gravitaional field inside the shell is zero, so no work required.</div>
source_db: MarksBatch2_P3.db
title:
there-is-a-shell-of-mass-m-and-density-of-the-shell-is-uniform-the-work-done-to-take-a-point-mass-from-point-a-to-b-is-ab-r-1

slug:
there-is-a-shell-of-mass-m-and-density-of-the-shell-is-uniform-the-work-done-to-take-a-point-mass-from-point-a-to-b-is-ab-r-1-13686

question:
<div class="question">There is a shell of mass $\mathrm{M}$ and density of the shell is uniform. The work done to take a point mass from point $\mathrm{A}$ to $\mathrm{B}$ is $[\mathrm{AB}=\mathrm{r}]$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/VKDS3exp_mPpaUxE5VNAVq47HazwfPBr5FF5wVCqd7o.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{GmM}}{\mathrm{r}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{GmM}}{\mathrm{R}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{\mathrm{GmM}}{\mathrm{r}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">zero</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">zero</span> </div>
answer: <div class="solution">Gravitaional field inside the shell is zero, so no work required.</div>
source_db: MarksBatch2_P3.db
title:
there-is-a-shell-of-mass-m-and-density-of-the-shell-is-uniform-the-work-done-to-take-a-point-mass-from-point-a-to-b-is-ab-r

slug:
there-is-a-shell-of-mass-m-and-density-of-the-shell-is-uniform-the-work-done-to-take-a-point-mass-from-point-a-to-b-is-ab-r-94236

question:
<div class="question">There is a shell of mass $\mathrm{M}$ and density of the shell is uniform. The work done to take a point mass from point $\mathrm{A}$ to $\mathrm{B}$ is $[\mathrm{AB}=\mathrm{r}]$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/VKDS3exp_mPpaUxE5VNAVq47HazwfPBr5FF5wVCqd7o.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{GmM}}{\mathrm{r}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{GmM}}{\mathrm{R}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{\mathrm{GmM}}{\mathrm{r}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">zero</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>


title: thermosetting-polymer-bakelite-is-formed-by-the-reaction-of-phenol-with
slug: thermosetting-polymer-bakelite-is-formed-by-the-reaction-of-phenol-with-23753
question: <div class="question">Thermosetting polymer bakelite is formed by the reaction of phenol with</div>
tags: ['Chemistry', 'Polymers', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mo>·</mo><msub><mi>CH</mi><mn>2</mn></msub><mo>·</mo><mi>CHO</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mo>.</mo><mi>CHO</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi><mo>.</mo><mi>CHO</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HCOOH</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi><mo>.</mo><mi>CHO</mi></math></span> </div>

answer:
<div class="solution"><p>Bakelite is a thermosetting plastic which is formed by the reaction of phenol with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">H</mi><mo>.</mo><mi>CHO</mi></math> in the presence of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><msub><mi>SO</mi><mn>4</mn></msub></math> (conc).</p><p>The reaction occurs as follows:</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2c748085-481d-4449-8dd7-151e7fcde4bf-image.png" style="width: 823px; height: 242px;"/></p><p>A cross-linked condensation polymer (at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">o</mi></math>-and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">p</mi></math>-position) bakelite is thus produced.</p></div>


tags: ['Physics', 'Kinetic Theory of Gases', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
title:
therms-velocity-of-oxygen-molecule-at-1-6-c-is-474-m-sec-the-rms-velocity-in-m-s-of-hydrogen-molecule-at-12-7-c-is

slug:
therms-velocity-of-oxygen-molecule-at-1-6-c-is-474-m-sec-the-rms-velocity-in-m-s-of-hydrogen-molecule-at-12-7-c-is-55296

question:
<div class="question">Ther.m.s. velocity of oxygen molecule at $16^{\circ} \mathrm{C}$ is $474 \mathrm{~m} / \mathrm{sec}$. The r.m.s. velocity in $\mathrm{m} / \mathrm{s}$ of hydrogen molecule at $127^{\circ} \mathrm{C}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1603</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1896</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">2230.59</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2730</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2230.59</span> </div>

answer:
<div class="solution">$v_{\operatorname{oxg} .}=\sqrt{\frac{3 R \times 289}{32}}\left(v_{\mathrm{rms}}=\sqrt{\frac{3 R T}{\mathrm{M}}}\right)$
<br/> <br/>$\mathrm{v}_{\mathrm{H}}=\sqrt{\frac{3 \mathrm{R} \times 400}{2}}$ so $\mathrm{v}_{\mathrm{H}}=2230.59 \mathrm{~m} / \mathrm{sec}$</div>


title: thiamine-and-pyridoxine-are-also-known-respectively-as-1
slug: thiamine-and-pyridoxine-are-also-known-respectively-as-1-36177
question: <div class="question">Thiamine and pyridoxine are also known respectively as :</div>
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
answer: <div class="solution">Vitamin $B_1$ - Thiamin Vitamin<br/>$\mathrm{B}_6$ - Pyridoxine</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>Vitamin $B_2$ and Vitamin $E$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>Vitamin $E$ and Vitamin $B_2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>Vitamin $B_6$ and Vitamin $B_2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>Vitamin $B_1$ and Vitamin $B_6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>Vitamin $B_1$ and Vitamin $B_6$</span> </div>


title: thomas-slag-is
slug: thomas-slag-is-94972
question: <div class="question">Thomas slag is</div>
tags: ['Chemistry', 's Block Elements', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Ca}_{3}\left(\mathrm{PO}_{4}\right)_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{CaSiO}_{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Mixture of (a) and (b)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{FeSiO}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Mixture of (a) and (b)</span> </div>

answer:
<div class="solution">Calcium silicophosphate (a mixture of $\left.\mathrm{Ca}_{3}\left(\mathrm{PO}_{4}\right)_{2} \&amp; \mathrm{Ca}_{2} \mathrm{SiO}_{4}\right)$ is called Thomas slag.</div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
title:
three-blocks-of-masses-m-1-m-2-and-m-3-are-connected-by-massless-strings-as-shown-on-a-frictionless-table-they-are-pulled-with-a-force-t-3-40-n-if-m-1

slug:
three-blocks-of-masses-m-1-m-2-and-m-3-are-connected-by-massless-strings-as-shown-on-a-frictionless-table-they-are-pulled-with-a-force-t-3-40-n-if-m-1-11537

question:
<div class="question">Three blocks of masses $\mathrm{m}_{1}, \mathrm{~m}_{2}$ and $\mathrm{m}_{3}$ are connected by massless strings, as shown, on a frictionless table. They are pulled with a force $\mathrm{T}_{3}$ $=40 \mathrm{~N}$. If $m_{1}=10 \mathrm{~kg}, \mathrm{~m}_{2}=6 \mathrm{~kg}$ and $\mathrm{m}_{3}=4 \mathrm{~kg},$ the<br/>tension $\mathrm{T}_{2}$ will be<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ZlQID5EcMiOmt14P9NLA3jntPiW4zJgXRLqAZ2hfr0Q.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20 \mathrm{~N}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$40 \mathrm{~N}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \mathrm{~N}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$32 \mathrm{~N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$32 \mathrm{~N}$</span> </div>

answer:
<div class="solution">For equilibrium of all 3 masses,<br/>$$<br/>a=\frac{\mathrm{T}_{3}}{\mathrm{~m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}}<br/>$$<br/>For equilibrium of $\mathrm{m}_{1} \&amp; \mathrm{~m}_{2}$ $\mathrm{T}_{2}=\left(\mathrm{m}_{1}+\mathrm{m}_{2}\right) \cdot \mathrm{a}$ or, $\mathrm{T}_{2}=\frac{\left(\mathrm{m}_{1}+\mathrm{m}_{2}\right) \mathrm{T}_{3}}{\mathrm{~m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}}$<br/>Given $m_{1}=10 \mathrm{~kg}, \mathrm{~m}_{2}=6 \mathrm{~kg}, \mathrm{~m}_{3}=4 \mathrm{~kg}$<br/>$\mathrm{T}_{3}=40 \mathrm{~N}$<br/>$\therefore \mathrm{T}_{2}=\frac{(10+6) \cdot 40}{10+6+4}=32 \mathrm{~N}$</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
title:
three-bricks-each-of-length-l-and-mass-m-are-arranged-as-shown-from-the-wall-the-distance-of-the-centre-of-mass-of-the-system-from-the-wall-is

slug:
three-bricks-each-of-length-l-and-mass-m-are-arranged-as-shown-from-the-wall-the-distance-of-the-centre-of-mass-of-the-system-from-the-wall-is-40086

question:
<div class="question">Three bricks each of length $L$ and mass $M$ are arranged as shown from the wall. The distance of the centre of mass of the system from the wall is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/qMdxJfwjUdkLzDUsPKM0YdQXmw-xVZisZyfqigQuYY8.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{L}}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{L}}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{3}{2}\right) \mathrm{L}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\left(\frac{11}{12}\right) \mathrm{L}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{11}{12}\right) \mathrm{L}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/u7DJxTmuY-PRriQ_WqzP3NpTSxNbz31Yer_aINc-e1s.original.fullsize.png"/><br/> <br/> <br/>$x_{1}=\frac{\mathrm{L}}{2}, x_{2}=\mathrm{L}, x_{3}=\frac{5 \mathrm{~L}}{4}$
<br/> <br/>$\therefore x_{\mathrm{CM}}=\frac{m_{1} x_{1}+m_{2} x_{2}+m_{3} x_{3}}{m_{1}+m_{2}+m_{3}}$
<br/> <br/>$=\frac{\mathrm{M} \times \frac{\mathrm{L}}{2}+\mathrm{M} \times \mathrm{L}+\mathrm{M} \times \frac{5 \mathrm{~L}}{4}}{\mathrm{M}+\mathrm{M}+\mathrm{M}}=\frac{11 \mathrm{~L}}{12}$</div>


tags: ['Physics', 'Capacitance', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
three-capacitors-c-1-1-f-c-2-2-f-and-c-3-3-f-are-connected-as-shown-in-figure-then-the-equivalent-capacitance-between-points-a-and-b-is-1

slug:
three-capacitors-c-1-1-f-c-2-2-f-and-c-3-3-f-are-connected-as-shown-in-figure-then-the-equivalent-capacitance-between-points-a-and-b-is-1-59100

question:
<div class="question">Three capacitors $\mathrm{C}_{1}=1 \mu \mathrm{F}, \mathrm{C}_{2}=2 \mu \mathrm{F}$ and $\mathrm{C}_{3}=3$
<br/> <br/>$\mu \mathrm{F}$ are connected as shown in figure, then the equivalent capacitance between points A and $\mathrm{B}$ is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/nKt5fLXSPwjtsXbOg4wO8EatBqST2InQ8Z6UkJFzx0U.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 \mu \mathrm{F}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \mu \mathrm{F}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 \mu \mathrm{F}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$6 \mu \mathrm{F}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6 \mu \mathrm{F}$</span> </div>

answer:
<div class="solution">$C_{\mathrm{eq}}=(1+2+3) \mu \mathrm{F}=6 \mu \mathrm{F}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/T3_BJa5_QUYp1OR9JBd76KOMdM-IxehcKrjQY5bDF_U.original.fullsize.png"/></div>


tags: ['Physics', 'Capacitance', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
three-capacitors-c-1-1-f-c-2-2-f-and-c-3-3-f-are-connected-as-shown-in-figure-then-the-equivalent-capacitance-between-points-a-and-b-is

slug:
three-capacitors-c-1-1-f-c-2-2-f-and-c-3-3-f-are-connected-as-shown-in-figure-then-the-equivalent-capacitance-between-points-a-and-b-is-79098

question:
<div class="question">Three capacitors $\mathrm{C}_{1}=1 \mu \mathrm{F}, \mathrm{C}_{2}=2 \mu \mathrm{F}$ and $\mathrm{C}_{3}=3$
<br/> <br/>$\mu \mathrm{F}$ are connected as shown in figure, then the equivalent capacitance between points A and $\mathrm{B}$ is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/nKt5fLXSPwjtsXbOg4wO8EatBqST2InQ8Z6UkJFzx0U.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 \mu \mathrm{F}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \mu \mathrm{F}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 \mu \mathrm{F}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$6 \mu \mathrm{F}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6 \mu \mathrm{F}$</span> </div>

answer:
<div class="solution">$C_{\mathrm{eq}}=(1+2+3) \mu \mathrm{F}=6 \mu \mathrm{F}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/T3_BJa5_QUYp1OR9JBd76KOMdM-IxehcKrjQY5bDF_U.original.fullsize.png"/></div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
title:
three-charge-q-q-and-4q-are-placed-in-a-straight-line-of-length-l-at-points-distant-0-2-1-and-l-respectively-from-one-end-in-order-to-make-the-net-fro

slug:
three-charge-q-q-and-4q-are-placed-in-a-straight-line-of-length-l-at-points-distant-0-2-1-and-l-respectively-from-one-end-in-order-to-make-the-net-fro-24155

question:
<div class="question">Three charge q, Q and 4q are placed in a straight line of length $l$ at points distant $0, \frac{1}{2}$ and $l$ respectively from one end. In order to make the net froce on q zero, the charge Q must be equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\mathrm{q}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-2 q$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{-\mathrm{q}}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">q</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\mathrm{q}$</span> </div>

answer:
<div class="solution">$\left(\mathrm{F}_{\text {net }}\right)_{\mathrm{q}}=0$
<br/> <br/>$\Rightarrow \quad \mathrm{k} \frac{\mathrm{Qq}}{\left(\frac{\ell}{2}\right)^{2}}+\mathrm{k} \frac{4 \mathrm{q}^{2}}{\ell^{2}}=0$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ZvOvFUXON_xQCFHF2M09zraYh8gPpGX6EQ5cMMGa20k.original.fullsize.png"/><br/> <br/> <br/>$\begin{array}{l}
<br/> <br/>\text { where } \mathrm{k}=\frac{1}{4 \pi \varepsilon_{0}} \\
<br/> <br/>\Rightarrow 4 \mathrm{Qq}+4 \mathrm{q}^{2}=0 \quad \Rightarrow \mathrm{Q}=-\mathrm{q}
<br/> <br/>\end{array}$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">zero</span> </div>
source_db: MarksBatch2_P3.db
title:
three-charges-are-placed-at-the-three-vertices-of-an-equilateral-triangle-of-side-a-as-shown-in-the-figure-the-force-experienced-by-the-charge-placed-

slug:
three-charges-are-placed-at-the-three-vertices-of-an-equilateral-triangle-of-side-a-as-shown-in-the-figure-the-force-experienced-by-the-charge-placed-34463

question:
<div class="question"><p>Three charges are placed at the three vertices of an equilateral triangle of side <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> as shown in the figure. The force experienced by the charge placed at the vertex <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> in a direction normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/8e9baabd-eb7a-44a9-8fdc-f4f176b3ecf0-image.png" style="width: 240px; height: 185px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>Q</mi><mn>2</mn></msup><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><msup><mi>Q</mi><mn>2</mn></msup><mrow><mn>4</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">zero</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>Q</mi><mn>2</mn></msup><mrow><mn>2</mn><mi>π</mi><msub><mi>ε</mi><mn>0</mn></msub><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></math></span> </li> </ul>

answer:
<div class="solution"><p>Resultant force</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>F</mi><mo>'</mo></msup><mo>=</mo><msqrt><msup><mi>F</mi><mn>2</mn></msup><mo>+</mo><msup><mi>F</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>F</mi><mi>F</mi><mi>cos</mi><mn>120</mn><mo>°</mo></msqrt><mo>=</mo><mi>F</mi></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mi>cos</mi><mn>120</mn><mo>°</mo><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2ca33903-5b85-4ca6-a0d8-ece8058f4f1e-image.png" style="width: 250px; height: 220px;"/></p><p>Now, from figure</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><msqrt><msup><mi>F</mi><mrow><mo>'</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>F</mi><mrow><mo>'</mo><mo>'</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><msup><mi>F</mi><mo>'</mo></msup><msup><mi>F</mi><mrow><mo>'</mo><mo>'</mo></mrow></msup><mi>cos</mi><mn>90</mn><mo>°</mo></msqrt></math></p><p>Now, the force normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> at vertex <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>F</mi><mrow><mo>'</mo><mo>'</mo></mrow></msup><mo>=</mo><msqrt><msup><mi>F</mi><mn>2</mn></msup><mo>-</mo><msup><mi>F</mi><mrow><mo>'</mo><mn>2</mn></mrow></msup></msqrt><mo>=</mo><mn>0</mn></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>∵</mo><msup><mi>F</mi><mo>'</mo></msup><mo>=</mo><mi>F</mi></mrow></mfenced></math></p></div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
three-equal-charges-q-are-placed-at-corners-of-an-equilateral-triangle-of-side-a-the-force-on-any-charge-is

slug:
three-equal-charges-q-are-placed-at-corners-of-an-equilateral-triangle-of-side-a-the-force-on-any-charge-is-46498

question:
<div class="question">Three equal charges (q) are placed at corners of an equilateral triangle of side $a$. The force on any charge is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">zero</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\sqrt{3} \frac{\mathrm{Kq}^{2}}{\mathrm{a}^{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{Kq}^{2}}{\sqrt{3} \mathrm{a}^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \sqrt{3} \frac{\mathrm{Kq}^{2}}{\mathrm{a}^{2}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{3} \frac{\mathrm{Kq}^{2}}{\mathrm{a}^{2}}$</span> </div>

answer:
<div class="solution">$\mathrm{F}_{\text {net }}=\sqrt{\mathrm{F}^{2}+\mathrm{F}^{2}+2 \mathrm{~F}^{2} \cos 60^{\circ}}=\sqrt{3} \mathrm{~F}$</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
title:
three-numbers-are-chosen-at-random-without-replacement-from-1-2-3-8-the-probability-that-their-minimum-is-3-given-that-their-maximum-is-6-is-1

slug:
three-numbers-are-chosen-at-random-without-replacement-from-1-2-3-8-the-probability-that-their-minimum-is-3-given-that-their-maximum-is-6-is-1-95561

question:
<div class="question">Three numbers are chosen at random without replacement from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>…</mo><mo>,</mo><mn>8</mn><mo>}</mo></math>. The probability that their minimum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>, given that their maximum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>8</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>5</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>5</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>5</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> be the event that maximum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> be the event that minimum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mi>A</mi></mfenced><mo>=</mo><mfrac><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>5</mn></mmultiscripts><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>8</mn></mmultiscripts></mfrac></math> (number less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mi>B</mi></mfenced><mo>=</mo><mfrac><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>5</mn></mmultiscripts><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>8</mn></mmultiscripts></mfrac></math> (number greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow></mfenced><mo>=</mo><mfrac><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>2</mn></mmultiscripts><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>8</mn></mmultiscripts></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Required probability</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>P</mi><mfenced><mrow><mi>B</mi><mo>/</mo><mi>A</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>A</mi><mo>∩</mo><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced><mi>A</mi></mfenced></mrow></mfrac><mo>=</mo><mfrac><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>2</mn></mmultiscripts><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>5</mn></mmultiscripts></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>10</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac></math></p></div>


tags: ['Physics', 'Rotational Motion', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
title:
three-particles-each-of-mass-m-gram-are-situated-at-the-vertices-of-an-equilateral-triangle-a-bc-of-side-cm-as-shown-in-the-figure-the-moment-of-inert

slug:
three-particles-each-of-mass-m-gram-are-situated-at-the-vertices-of-an-equilateral-triangle-a-bc-of-side-cm-as-shown-in-the-figure-the-moment-of-inert-61986

question:
<div class="question">Three particles, each of mass $m$ gram, are situated at the vertices of an equilateral triangle $A B C$ of side $\ell \mathrm{~cm}$ (as shown in the figure). The moment of inertia of the system about a line $A X$ perpendicular to $A B$ and in the plane of $A B C$, in gram $-\mathrm{cm}^2$ units will be
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2za3QMTRGZ5dV8Nb4P_9Rtw3Ka8TeXr4HDIyaJHK_U0.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{2} m l^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3}{4} m l^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 m l^{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{5}{4} m l^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{4} m l^{2}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/lGO2-PSVc5elOdsJMF-jQx9Acl8-LXMD7nj3eFgXusE.original.fullsize.png"/><br/> <br/> <br/>$\mathrm{I}_{\mathrm{AX}}=\mathrm{m}(\mathrm{AB})^{2}+\mathrm{m}(\mathrm{OC})^{2}=\mathrm{m} l^{2}+\mathrm{m}\left(l \cos 60^{\circ}\right)^{2}=m l^{2}+\frac{\mathrm{m} l^{2}}{4}=\frac{5}{4} \mathrm{~m} l^{2}$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
title:
three-rods-of-identical-crosssectional-area-and-made-from-the-same-metal-form-the-sides-of-an-isosceles-triangle-a-b-c-right-angled-at-b-as-shown-in-f

slug:
three-rods-of-identical-crosssectional-area-and-made-from-the-same-metal-form-the-sides-of-an-isosceles-triangle-a-b-c-right-angled-at-b-as-shown-in-f-75240

question:
<div class="question">Three rods of identical cross-sectional area and made from the same metal, form the sides of an isosceles triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi></math> right angled at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> as shown in figure. The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are maintained at temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mi>T</mi></math> respectively, in the steady state. Now, assuming that only heat conduction takes place. The temperature of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> will be<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/902d5ed5-7c34-42f9-b93b-f5e60dd206eb-image.png" style="width: 200px; height: 188px;"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>T</mi><mrow><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>T</mi><mrow><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>1</mn></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msqrt><mn>3</mn></msqrt><mi>T</mi></mrow><mrow><mo>(</mo><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>0</mn></msub></math> be the temperature of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> be the length of rod <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math>.</p><p>Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>A</mi><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>x</mi></math></p><p>At steady state, the rate of heat flowing from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo></math> rate of heat flowing from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p><p>So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>K</mi><mi>A</mi><mfenced><mrow><msqrt><mn>2</mn></msqrt><mi>T</mi><mo>-</mo><msub><mi>T</mi><mn>0</mn></msub></mrow></mfenced></mrow><mi>x</mi></mfrac><mo>=</mo><mfrac><mrow><mi>K</mi><mi>A</mi><mfenced><mrow><msub><mi>T</mi><mn>0</mn></msub><mo>-</mo><mi>T</mi></mrow></mfenced></mrow><mrow><msqrt><mn>2</mn></msqrt><mi>x</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mfenced><mrow><msqrt><mn>2</mn></msqrt><mi>T</mi><mo>-</mo><msub><mi>T</mi><mn>0</mn></msub></mrow></mfenced><mo>=</mo><msub><mi>T</mi><mn>0</mn></msub><mo>-</mo><mi>T</mi></math></p><p>By solving, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>0</mn></msub><mo>=</mo><mfrac><mrow><mn>3</mn><mi>T</mi></mrow><mrow><mo>(</mo><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac></math></p></div>


tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
three-separate-samples-of-a-solution-of-a-single-salt-gave-these-results-one-formed-a-white-precipitate-with-excess-ammonia-solution-one-formed-a-whit

slug:
three-separate-samples-of-a-solution-of-a-single-salt-gave-these-results-one-formed-a-white-precipitate-with-excess-ammonia-solution-one-formed-a-whit-67203

question:
<div class="question">Three separate samples of a solution of a single salt gave these results. One formed a white precipitate with excess ammonia solution, one formed a white precipitate with dil. HCl solution and one formed a black precipitate with $\mathrm{H}_{2} \mathrm{~S}$. The salt could be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{AgNO}_{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Hg}\left(\mathrm{NO}_{3}\right)_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{MnSO}_{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}$</span> </div>

answer:
<div class="solution">$\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}+2 \mathrm{NH}_{4} \mathrm{OH}$<br/>$$<br/>\begin{array}{r}<br/>\rightarrow \mathrm{Pb}(\mathrm{OH})_{2} \downarrow+2 \mathrm{NH}_{4} \mathrm{NO}_{3}(\text { white ppt }) \\<br/>\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}+2 \mathrm{HCl} \rightarrow \mathrm{PbCl}_{2} \downarrow+2 \mathrm{HNO}_{3}(\text { whiteppt }) \\<br/>\mathrm{Pb}\left(\mathrm{NO}_{3}\right)_{2}+\mathrm{H}_{2} \mathrm{~S} \rightarrow \mathrm{PbS} \downarrow+2 \mathrm{HNO}_{3} \text { (black) }<br/>\end{array}<br/>$$</div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">A</span> </div>
source_db: MarksBatch2_P3.db
title:
three-specimens-a-b-c-of-same-radioactive-element-has-activities-1-microcurie-1-rutherford-and-1-becquerel-respectively-which-specimen-has-maximum-mas

slug:
three-specimens-a-b-c-of-same-radioactive-element-has-activities-1-microcurie-1-rutherford-and-1-becquerel-respectively-which-specimen-has-maximum-mas-66480

question:
<div class="question">Three specimens A, B, C of same radioactive element has activities 1 microcurie, 1 rutherford and 1 becquerel respectively. Which specimen has maximum mass?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">A</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">B</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">C</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">all have equal masses</span> </li> </ul>

answer:
<div class="solution">1 becquerel $=1 \mathrm{dis} / \mathrm{s}$
<br/> <br/>1 rutherford $=10^{6} \mathrm{dis} / \mathrm{s}$
<br/> <br/>1 curie $=3.7 \times 10^{10} \mathrm{dis} / \mathrm{s}$</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
three-voltmeters-a-b-and-c-having-resistances-r-15-r-and-3-r-respectively-are-connected-as-shown-when-some-potential-difference-is-applied-between-x-a-1

slug:
three-voltmeters-a-b-and-c-having-resistances-r-15-r-and-3-r-respectively-are-connected-as-shown-when-some-potential-difference-is-applied-between-x-a-1-97853

question:
<div class="question">Three voltmeters $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ having resistances $\mathrm{R}, 1.5 \mathrm{R}$ and $3 \mathrm{R},$ respectively, are connected as shown. When some potential difference is applied between $\mathrm{X}$ and $\mathrm{Y},$ the voltmeter readings are $\mathrm{V}_{\mathrm{A}}, \mathrm{V}_{\mathrm{B}}$ and $\mathrm{V}_{\mathrm{C}}$ respectively. Then -
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PQtngVEcNnNdrAgkGVfNw8qCVL1DFPZFfmEiycMN1Xo.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$V_{A} \neq V_{B}=V_{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$V_{\mathrm{A}}=V_{\mathrm{B}} \neq V_{\mathrm{C}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}} \neq \mathrm{V}_{\mathrm{C}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$V_{A}=V_{B}=V_{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$V_{A}=V_{B}=V_{C}$</span> </div>

answer:
<div class="solution">$V_{A}=I R$
<br/>$V_{B}=\left(\frac{2 I}{3}\right) 1.5 R=I R \quad V_{C}=\left(\frac{1}{3}\right) 3 R=I R$
<br/>$\therefore \quad V_{A}=V_{B}=V_{C}$</div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
title:
three-voltmeters-a-b-and-c-having-resistances-r-15-r-and-3-r-respectively-are-connected-as-shown-when-some-potential-difference-is-applied-between-x-a

slug:
three-voltmeters-a-b-and-c-having-resistances-r-15-r-and-3-r-respectively-are-connected-as-shown-when-some-potential-difference-is-applied-between-x-a-16139

question:
<div class="question">Three voltmeters $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ having resistances $\mathrm{R}, 1.5 \mathrm{R}$ and $3 \mathrm{R},$ respectively, are connected as shown. When some potential difference is applied between $\mathrm{X}$ and $\mathrm{Y},$ the voltmeter readings are $\mathrm{V}_{\mathrm{A}}, \mathrm{V}_{\mathrm{B}}$ and $\mathrm{V}_{\mathrm{C}}$ respectively. Then -
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PQtngVEcNnNdrAgkGVfNw8qCVL1DFPZFfmEiycMN1Xo.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$V_{A} \neq V_{B}=V_{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$V_{\mathrm{A}}=V_{\mathrm{B}} \neq V_{\mathrm{C}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{V}_{\mathrm{A}} \neq \mathrm{V}_{\mathrm{B}} \neq \mathrm{V}_{\mathrm{C}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$V_{A}=V_{B}=V_{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$V_{A}=V_{B}=V_{C}$</span> </div>

answer:
<div class="solution">$V_{A}=I R$
<br/>$V_{B}=\left(\frac{2 I}{3}\right) 1.5 R=I R \quad V_{C}=\left(\frac{1}{3}\right) 3 R=I R$
<br/>$\therefore \quad V_{A}=V_{B}=V_{C}$</div>


tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
through-the-vertex-o-of-a-parabola-y-2-4-x-chords-op-and-oq-are-drawn-at-right-angles-to-one-another-the-locus-of-the-middle-point-of-pq-is

slug:
through-the-vertex-o-of-a-parabola-y-2-4-x-chords-op-and-oq-are-drawn-at-right-angles-to-one-another-the-locus-of-the-middle-point-of-pq-is-39326

question:
<div class="question">Through the vertex $\mathrm{O}$ of a parabola $\mathrm{y}^{2}=4 \mathrm{x}$, chords OP and $O Q$ are drawn at right angles to one another. The locus of the middle point of $\mathrm{PQ}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y^{2}=2 x+8$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad y^{2}=x+8$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y^{2}=2 x-8$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad y^{2}=x-8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y^{2}=2 x-8$</span> </div>

answer:
<div class="solution">Given parabola is $\mathrm{y}^{2}=4 ...(1)
<br/> <br/>\mathrm{x}$ Let $\mathrm{P} \equiv\left(\mathrm{t}_{1}^{2}, 2 \mathrm{t}_{1}\right)$ and $\mathrm{Q} \equiv\left(\mathrm{t}_{2}^{2}, 2 \mathrm{t}_{2}\right)$
<br/> <br/>Slope of $\mathrm{OP}=\frac{2 \mathrm{t}_{1}}{\mathrm{t}_{1}^{2}}=\frac{2}{\mathrm{t}_{1}}$ and slope of $\mathrm{OQ}=\frac{2}{\mathrm{t}_{2}}$
<br/> <br/>Since $\mathrm{OP} \perp \mathrm{OQ}, \therefore \frac{4}{\mathrm{t}_{1} \mathrm{t}_{2}}=-1$ or $\mathrm{t}_{1} \mathrm{t}_{2}=-4 \ldots(2)$
<br/> <br/>Let $\mathrm{R}(\mathrm{h}, \mathrm{k})$ be the middle point of $\mathrm{PQ},$ then $\mathrm{h}=\frac{\mathrm{t}_{1}^{2}+\mathrm{t}_{2}^{2}}{2} \quad \ldots(3) \quad$ and $\quad \mathrm{k}=\mathrm{t}_{1}+\mathrm{t}_{2} \ldots(4)$
<br/> <br/>From $(4), \mathrm{k}^{2}=\mathrm{t}_{1}^{2}+\mathrm{t}_{2}^{2}+2 \mathrm{t}_{1} \mathrm{t}_{2}=2 \mathrm{~h}-8 \quad[\mathrm{From}$
<br/> <br/>(2) and (3)] Hence locus of $\mathrm{R}(\mathrm{h}, \mathrm{k})$ is $\mathrm{y}^{2}=2 \mathrm{x}-8$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
title:
to-increase-the-current-sensitivity-of-a-moving-coil-galvanometer-by-50-its-resistance-is-increased-so-that-new-resistance-becomes-twice-its-initial-r

slug:
to-increase-the-current-sensitivity-of-a-moving-coil-galvanometer-by-50-its-resistance-is-increased-so-that-new-resistance-becomes-twice-its-initial-r-10354

question:
<div class="question">To increase the current sensitivity of a moving coil galvanometer by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo> </mo><mo>%</mo><mo>,</mo></math> its resistance is increased, so that new resistance becomes twice its initial resistance, by what factor does its voltage sensitivity change?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Increases by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mo>%</mo></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Decreases by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mo>%</mo></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Increases by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mo>%</mo></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Decreases by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mo>%</mo></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Decreases by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mo>%</mo></math></span> </div>

answer:
<div class="solution"><p>New, current sensitivity, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>I</mi><mi>s</mi><mo>'</mo></msubsup><mo>=</mo><msub><mi>I</mi><mi>s</mi></msub><mo>+</mo><mfrac><mn>50</mn><mn>100</mn></mfrac><msub><mi>I</mi><mi>s</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>150</mn><mn>100</mn></mfrac><msub><mi>I</mi><mi>s</mi></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msub><mi>I</mi><mi>s</mi></msub></math></p><p>New resistance, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>R</mi><mo>'</mo></msup><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">R</mi></math></p><p>Initial voltage sensitivity, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mfrac><msub><mi>I</mi><mi>s</mi></msub><mi>R</mi></mfrac></math></p><p>Now, new voltage sensitivity.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mfrac><msubsup><mi>i</mi><mi>s</mi><mo>'</mo></msubsup><msup><mi>R</mi><mo>'</mo></msup></mfrac><mo>=</mo><mfrac><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo> </mo><msub><mi>I</mi><mi>c</mi></msub></mrow><mrow><mn>2</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow></mfrac><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><msub><mi>V</mi><mi>s</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>%</mo></math> decrease in voltage sensitivity</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mi>s</mi></msub><mo>-</mo><msubsup><mi>V</mi><mi>s</mi><mo>'</mo></msubsup></mrow><msub><mi>V</mi><mi>s</mi></msub></mfrac><mo>×</mo><mn>100</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><msub><mi>V</mi><mi>s</mi></msub><msub><mi>V</mi><mi>s</mi></msub></mfrac></mrow></mfenced><mo>×</mo><mn>100</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced><mo>×</mo><mn>100</mn><mo>=</mo><mn>25</mn><mo> </mo><mo>%</mo></math></p></div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">400</span> </div>
source_db: MarksBatch2_P3.db
title:
total-number-of-ways-of-selecting-2-white-squares-on-a-normal-chess-board-if-the-squares-are-not-from-the-same-row-or-column-is-equal-to

slug:
total-number-of-ways-of-selecting-2-white-squares-on-a-normal-chess-board-if-the-squares-are-not-from-the-same-row-or-column-is-equal-to-66052

question:
<div class="question">Total number of ways of selecting 2 white squares on a normal chess board if the squares are not from the same row or column is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">480</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">496</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">412</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">400</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">2 white squares can be selected in ${ }^{32} \mathrm{C}_{2}$ ways If they belong to some row or same column then $2\left(8 \cdot{ }^{4} \mathrm{C}_{2}\right)$ ways.</div>


tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
transfer-characteristics-output-voltage-v-0-vs-input-voltage-v-1-for-a-base-biased-transistor-in-ce-configuration-is-as-shown-in-the-figure-for-using-

slug:
transfer-characteristics-output-voltage-v-0-vs-input-voltage-v-1-for-a-base-biased-transistor-in-ce-configuration-is-as-shown-in-the-figure-for-using-95375

question:
<div class="question">Transfer characteristics [output voltage $\left(V_{0}\right)$ vs input voltage $\left.\left(V_{1}\right)\right]$ for a base biased transistor in CE configuration is as shown in the figure. For using transistor as a switch, it is used
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/enat0to2WZMFnv9meOLa6dTms7I21pTlGlYM1CbLqGs.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">in region (III)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">both in region (I) and (II)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">in region (II)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">in region (I)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">both in region (I) and (II)</span> </div>

answer:
<div class="solution">$\mathrm{I} \rightarrow \mathrm{ON}$
<br/> <br/>$\mathrm{II} \rightarrow \mathrm{OFF}$
<br/> <br/>In $I^{\text {nd }}$ state it is used as a amplifier it is active region.</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
title:
transverse-waves-of-the-same-frequency-are-generated-in-two-steel-wires-a-and-b-the-diameter-of-a-is-twice-that-of-b-and-the-tension-in-a-is-half-that

slug:
transverse-waves-of-the-same-frequency-are-generated-in-two-steel-wires-a-and-b-the-diameter-of-a-is-twice-that-of-b-and-the-tension-in-a-is-half-that-27280

question:
<div class="question">Transverse waves of the same frequency are generated in two steel wires $A$ and $B$. The diameter of $A$ is twice that of $B$ and the tension in $A$ is half that in $B$. The ratio of the velocities of the waves in $A$ and $B$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$1: 2$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$1: \sqrt{2}$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$1: 2 \sqrt{2}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$3: 2 \sqrt{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$1: 2 \sqrt{2}$
<br/></span> </div>

answer:
<div class="solution">The velocity of transverse waves is given by $v=\sqrt{\frac{T}{m}}$ where, $T=$ Tension and $m=$ mass per unit length of the wire. If $r$ is the radius of the wire and $\rho$ its density then,
<br/>$m=\pi r^2 \rho$
<br/>$\therefore \quad v=\sqrt{\frac{T}{m}}=\sqrt{\frac{T}{\pi r^2 \rho}}$
<br/>$\therefore \quad v_A=\frac{\sqrt{T_A}}{r_A \sqrt{\pi \rho}}$
<br/>and $\quad v_B=\frac{\sqrt{T_B}}{r_B \sqrt{\pi \rho}}$
<br/>Now, $\quad \frac{v_A}{v_B}=\sqrt{\frac{T_A}{T_B}} \cdot \frac{r_B}{r_A}$
<br/>$\because \quad r_A=2 r_B$ and $T_A=\frac{1}{2} T_B$
<br/>Hence, $\quad \frac{v_A}{v_B}=\frac{1}{2 \sqrt{2}}$</div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
two-balloons-are-filled-one-with-pure-he-gas-and-other-with-air-respectively-if-the-pressure-and-temperature-of-these-balloons-are-same-then-the-numbe

slug:
two-balloons-are-filled-one-with-pure-he-gas-and-other-with-air-respectively-if-the-pressure-and-temperature-of-these-balloons-are-same-then-the-numbe-44190

question:
<div class="question">Two balloons are filled one with pure he gas and other with air respectively. If the pressure and temperature of these balloons are same, then the number of molecules per unit volume is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">more in He gas filled balloon</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">same in both balloons</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">more in air filled balloon</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">in the ratio 1: 4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">same in both balloons</span> </div>

answer:
<div class="solution">Assuming the balloons have the same volume, as $\mathrm{pV}=\mathrm{nRT}$. If $\mathrm{P}, \mathrm{V}$ and $\mathrm{T}$ are the same, $\mathrm{n}$ the number of moles present will be the same, whether it is He or air. Hence, number of molecules per unit volume will be same in both the balloons.</div>


tags: ['Physics', 'Laws of Motion', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
two-blocks-a-and-b-are-placed-one-over-the-other-on-a-smooth-horizontal-surface-the-maximum-horizontal-force-that-can-be-applied-on-lower-block-a-so-t-1

slug:
two-blocks-a-and-b-are-placed-one-over-the-other-on-a-smooth-horizontal-surface-the-maximum-horizontal-force-that-can-be-applied-on-lower-block-a-so-t-1-77542

question:
<div class="question"><p>Two blocks <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are placed one over the other on a smooth horizontal surface. The maximum horizontal force that can be applied on lower block <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> move without separation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn><mo> </mo><mi mathvariant="normal">N</mi></math>. The coefficient of friction between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/036eec6b-edfd-4997-848c-e045e6d3ae84-image.png" style="width: 300px; height: 115px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>3</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math></span> </div>

answer:
<div class="solution"><p>Equation of motion of lower block is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mi>μ</mi><mfenced><mrow><msub><mi>M</mi><mi>A</mi></msub><mo>+</mo><msub><mi>M</mi><mi>B</mi></msub></mrow></mfenced><mo>·</mo><mi>g</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mi>F</mi><mrow><mfenced><mrow><msub><mi>M</mi><mi>A</mi></msub><mo>+</mo><msub><mi>M</mi><mi>B</mi></msub></mrow></mfenced><mi>g</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>49</mn><mrow><mn>10</mn><mo>×</mo><mn>9</mn><mo>.</mo><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math></p></div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
title:
two-blocks-a-and-b-are-placed-one-over-the-other-on-a-smooth-horizontal-surface-the-maximum-horizontal-force-that-can-be-applied-on-lower-block-a-so-t

slug:
two-blocks-a-and-b-are-placed-one-over-the-other-on-a-smooth-horizontal-surface-the-maximum-horizontal-force-that-can-be-applied-on-lower-block-a-so-t-37281

question:
<div class="question"><p>Two blocks <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> are placed one over the other on a smooth horizontal surface. The maximum horizontal force that can be applied on lower block <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> move without separation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>49</mn><mo> </mo><mi mathvariant="normal">N</mi></math>. The coefficient of friction between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/036eec6b-edfd-4997-848c-e045e6d3ae84-image.png" style="width: 300px; height: 115px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>3</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math></span> </div>

answer:
<div class="solution"><p>Equation of motion of lower block is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mi>μ</mi><mfenced><mrow><msub><mi>M</mi><mi>A</mi></msub><mo>+</mo><msub><mi>M</mi><mi>B</mi></msub></mrow></mfenced><mo>·</mo><mi>g</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mi>F</mi><mrow><mfenced><mrow><msub><mi>M</mi><mi>A</mi></msub><mo>+</mo><msub><mi>M</mi><mi>B</mi></msub></mrow></mfenced><mi>g</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>49</mn><mrow><mn>10</mn><mo>×</mo><mn>9</mn><mo>.</mo><mn>8</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math></p></div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3.3</span> </div>
source_db: MarksBatch2_P3.db
title:
two-blocks-are-connected-over-a-massless-pulley-as-shown-in-fig-the-mass-of-block-a-is-10-kg-and-the-coefficient-of-kinetic-friction-is-02-block-a-sli

slug:
two-blocks-are-connected-over-a-massless-pulley-as-shown-in-fig-the-mass-of-block-a-is-10-kg-and-the-coefficient-of-kinetic-friction-is-02-block-a-sli-61211

question:
<div class="question">Two blocks are connected over a massless pulley as shown in fig. The mass of block $A$ is $10 \mathrm{~kg}$ and the coefficient of kinetic<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/w26kFtINGpnHtEs9gsVBYOEWI7HlKLYwzUXNTe7-Hno.original.fullsize.png"/><br/><br/>friction is 0.2 . Block A slides down the incline at constant speed. The mass of block $\mathrm{B}$ in $\mathrm{kg}$ is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3.5</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">3.3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2.5</span> </li> </ul>

answer:
<div class="solution">Considering the equilibrium of A, we get<br/>\begin{array}{l}<br/>10 \mathrm{a}=10 \mathrm{~g} \sin 30^{\circ}-\mathrm{T}-\mu \mathrm{N} \\<br/>\text { where } \mathrm{N}=10 \mathrm{~g} \cos 30^{\circ}<br/>\end{array}<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/7muBoXQvLcuz2GnCKEYXMZqq_aRLzIlJj6Owfbbw5Vw.original.fullsize.png"/><br/><br/>$\therefore 10 \mathrm{a}=\frac{10}{2} \mathrm{~g}-\mathrm{T}-\mu \times 10 \mathrm{~g} \cos 30^{\circ}$<br/>but $\mathrm{a}=0, \mathrm{~T}=\mathrm{m}_{\mathrm{B}} \mathrm{g}$<br/>$0=5 \mathrm{~g}-\mathrm{m}_{\mathrm{B}} \mathrm{g}-\frac{0.2 \sqrt{3}}{2} \times 10 \times \mathrm{g}$<br/>$\Rightarrow \mathrm{m}_{\mathrm{B}}=3.268 \approx 3.3 \mathrm{~kg}$</div>


source_db: MarksBatch2_P3.db
title:
two-bodies-are-having-kinetic-energies-in-the-ratio-16-9-if-they-have-same-linear-momentum-the-ratio-of-their-masses-respectively-is

slug:
two-bodies-are-having-kinetic-energies-in-the-ratio-16-9-if-they-have-same-linear-momentum-the-ratio-of-their-masses-respectively-is-21480

question:
<div class="question">Two bodies are having kinetic energies in the ratio $16: 9$. If they have same linear momentum, the ratio of their masses respectively is:</div>

tags:
['Physics', 'Center of Mass Momentum and Collision', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>4 : 3</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>3 : 4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>16 : 9</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/>9 : 16</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>9 : 16</span> </div>

answer:
<div class="solution"><br/>$\begin{aligned} &amp; \text { K.E. }=\frac{\mathrm{p}^2}{2 \mathrm{~m}} \\ &amp; \therefore \frac{\mathrm{K}_1}{\mathrm{~K}_2}=\frac{\mathrm{p}_1^2}{2 \mathrm{~m}_1} \times \frac{2 \mathrm{~m}_2}{\mathrm{p}_2^2}=\frac{\mathrm{m}_2}{\mathrm{~m}_1}=\frac{16}{9} \\ &amp; \therefore \frac{\mathrm{m}_1}{\mathrm{~m}_2}=\frac{9}{16}\end{aligned}$</div>


tags: ['Physics', 'Capacitance', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
title:
two-capacitors-c-1-and-c-2-are-charged-to-120-v-and-200-v-respectively-it-is-found-that-by-connecting-them-together-the-potential-on-each-one-can-be-m-1

slug:
two-capacitors-c-1-and-c-2-are-charged-to-120-v-and-200-v-respectively-it-is-found-that-by-connecting-them-together-the-potential-on-each-one-can-be-m-1-49712

question:
<div class="question">Two capacitors $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ are charged to $120 \mathrm{~V}$ and $200 \mathrm{~V}$ respectively. It is found that by connecting them together the potential on each one can be made zero. Then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 C_{1}=3 C_{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3 C_{1}=5 C_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 C_{1}+5 C_{2}=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$9 C_{1}=4 C_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 C_{1}=5 C_{2}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/jJt7_1JooJMVJ7r7l-Vi3gZJm26QwjPZ7jK8J2r7ps8.original.fullsize.png"/><br/> <br/>For potential to be made zero, after connection
<br/>$120 \mathrm{C}_{1}=200 \mathrm{C}_{2} \quad\left[\because C=\frac{q}{v}\right]$
<br/>$\Rightarrow 3 C_{1}=5 C_{2}$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
two-circular-plates-of-radius-5-cm-each-have-a-001-mm-thick-water-film-between-them-then-what-will-be-the-force-required-to-separate-these-plate-st-of

slug:
two-circular-plates-of-radius-5-cm-each-have-a-001-mm-thick-water-film-between-them-then-what-will-be-the-force-required-to-separate-these-plate-st-of-73804

question:
<div class="question">Two circular plates of radius $5 \mathrm{~cm}$ each, have a 0.01 mm thick water film between them. Then what will be the force required to separate these plate (S.T. of water $=73$ dyne/cm)?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$125 \mathrm{~N}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$95 \mathrm{~N}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$115 \mathrm{~N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$105 \mathrm{~N}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$115 \mathrm{~N}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \mathrm{F}=&amp; \frac{2 \mathrm{AT}}{\mathrm{d}}=\frac{2 \times \pi \times(0.05)^{2} \times 73 \times 10^{-3}}{0.01 \times 10^{-3}} \\=&amp; 36.5 \pi \approx 115 \text { newton } \end{aligned}$</div>


tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
two-coaxial-solenoids-are-made-by-winding-thin-insulated-wire-over-a-pipe-of-crosssectional-area-a-10-cm-2-and-length-20-cm-if-one-of-the-solenoid-has

slug:
two-coaxial-solenoids-are-made-by-winding-thin-insulated-wire-over-a-pipe-of-crosssectional-area-a-10-cm-2-and-length-20-cm-if-one-of-the-solenoid-has-95745

question:
<div class="question">Two coaxial solenoids are made by winding thin insulated wire over a pipe of cross-sectional area $\mathrm{A}=10 \mathrm{~cm}^{2}$ and length $=20 \mathrm{~cm} .$ If one of the solenoid has 300 turns and the other 400 turns, their mutual inductance is $\left(\mu_{0}=4 \pi \times 10^{-7} \mathrm{TmA}^{-1}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.4 \pi \times 10^{-5} \mathrm{H}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4.8 \pi \times 10^{-4} \mathrm{H}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4.8 \pi \times 10^{-5} \mathrm{H}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2.4 \pi \times 10^{-4} \mathrm{H}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.4 \pi \times 10^{-4} \mathrm{H}$</span> </div>

answer:
<div class="solution">$\begin{aligned} M &amp;=\frac{\mu_{0} \mathrm{~N}_{1} \mathrm{~N}_{2} \mathrm{~A}}{\ell} \\ &amp;=\frac{4 \pi \times 10^{-7} \times 300 \times 400 \times 100 \times 10^{-4}}{0.2} \\ &amp;=2.4 \pi \times 10^{-4} \mathrm{H} \end{aligned}$
<br/> <br/>#</div>


title: two-dice-are-thrown-together-4-times-the-probability-that-both-dice-will-show-same-numbers-twice-is
tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
slug:
two-dice-are-thrown-together-4-times-the-probability-that-both-dice-will-show-same-numbers-twice-is-18998

question:
<div class="question">Two dice are thrown together 4 times. The probability that both dice will show same numbers twice is -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{25}{36}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{25}{216}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{25}{216}$</span> </div>

answer:
<div class="solution">(c) The probability of showing same number by both dice $p=\frac{6}{36}=\frac{1}{6}$
<br/>In binomial distribution here $n=4, r=2, p=$
<br/>$\begin{array}{l}
<br/>\frac{1}{6}, \mathrm{q}=\frac{5}{6} \\
<br/>\text { req. probability }={ }^{n} C_{r} q^{n-r} p^{r}={ }^{4} C_{2}\left(\frac{5}{6}\right)^{2}\left(\frac{1}{6}\right)^{2} \\
<br/>=6\left(\frac{25}{36}\right)\left(\frac{1}{36}\right)=\frac{25}{216}
<br/>\end{array}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
title:
two-electrons-are-moving-with-equal-speed-of-5-10-5-m-s-1-parallel-to-each-other-the-ratio-of-the-electrostatic-to-the-magnetic-force-between-them-is

slug:
two-electrons-are-moving-with-equal-speed-of-5-10-5-m-s-1-parallel-to-each-other-the-ratio-of-the-electrostatic-to-the-magnetic-force-between-them-is-50336

question:
<div class="question">Two electrons are moving with equal speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> parallel to each other. The ratio of the electrostatic to the magnetic force between them is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>4</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Given, speed of electrons, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/baa17b67-197b-4978-aef5-170aa0137fd4-image.png" style="width: 140px; height: 198px;"/></p><p>Electrostatic force between the electrons,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>e</mi></msub><mo>=</mo><mi>k</mi><mo>·</mo><mfrac><msup><mi>e</mi><mn>2</mn></msup><msup><mi>r</mi><mn>2</mn></msup></mfrac><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>Magnetic field produced due to moving electron with velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math>,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mfrac><msub><mi>μ</mi><mn>0</mn></msub><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mo>·</mo><mfrac><mrow><mi>e</mi><mi>v</mi></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>Magnetic force between the electrons,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>m</mi></msub><mo>=</mo><mi>B</mi><mi>e</mi><mi>v</mi><mo>=</mo><mfrac><msub><mi>μ</mi><mn>0</mn></msub><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mo>·</mo><mfrac><mrow><mi>e</mi><mi>v</mi></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac><mo>·</mo><mi>e</mi><mi>v</mi></math>   (from Eq. (ii))</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>m</mi></msub><mo>=</mo><mfrac><msub><mi>μ</mi><mn>0</mn></msub><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mo>·</mo><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup><msup><mi>v</mi><mn>2</mn></msup></mrow><msup><mi>r</mi><mn>2</mn></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> From Eqs. (i) and (iii), we have</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>F</mi><mi>e</mi></msub><msub><mi>F</mi><mi>m</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mi>k</mi><msup><mi>e</mi><mn>2</mn></msup><mo>/</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><mfrac><mrow><msub><mi>μ</mi><mn>0</mn></msub><msup><mi>e</mi><mn>2</mn></msup><msup><mi>v</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mi>π</mi><mo>·</mo><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfrac><mo>=</mo><mfrac><mrow><mi>k</mi><mo>·</mo><mn>4</mn><mi>π</mi></mrow><mrow><msub><mi>μ</mi><mn>0</mn></msub><msup><mi>v</mi><mn>2</mn></msup></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>9</mn><mo>×</mo><msup><mn>10</mn><mn>9</mn></msup><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>.</mo><mn>14</mn></mrow><mrow><mn>4</mn><mi>π</mi><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>7</mn></mrow></msup><msup><mfenced><mrow><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup></mrow></mfenced><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>3</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup></math></p></div>


tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">ionic</span> </div>
source_db: MarksBatch2_P3.db
title:
two-elements-a-and-b-have-electronegativities-1-2-and-3-0-respectively-the-nature-of-bond-between-a-and-b-would-be

slug:
two-elements-a-and-b-have-electronegativities-1-2-and-3-0-respectively-the-nature-of-bond-between-a-and-b-would-be-85448

question:
<div class="question"><p>Two elements <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> have electronegativities <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>0</mn></math> respectively. The nature of bond between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> would be</p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">ionic</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">covalent</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">co-ordinate</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">metallic</span> </li> </ul>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo></math> Difference of electronegativity between <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">A</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">B</mi><mo>)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>.</mo><mn>0</mn><mo>-</mo><mn>1</mn><mo>.</mo><mn>2</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>8</mn></math> and, a bond having electronegativity difference greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>65</mn></math> is of ionic nature.</p><p>Hence, bond between <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">A</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">B</mi><mo>)</mo></math> would be of ionic nature. </p></div>


tags: ['Chemistry', 's Block Elements', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
title:
two-elements-a-and-b-have-similar-chemical-properties-they-dont-form-solid-hydrogencarbonates-but-react-with-nitrogen-to-form-nitrides-a-and-b-respect-1

slug:
two-elements-a-and-b-have-similar-chemical-properties-they-dont-form-solid-hydrogencarbonates-but-react-with-nitrogen-to-form-nitrides-a-and-b-respect-1-87562

question:
<div class="question">Two elements $A$ and $B$ have similar chemical properties. They don't form solid hydrogencarbonates, but react with nitrogen to form nitrides. A and B, respectively, are :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$Na$ and $Rb$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$Na$ and $Ca$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$Cs$ and $Ba$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$Li$ and $Mg$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$Li$ and $Mg$</span> </div>

answer:
<div class="solution">Li and $\mathrm{Mg}$ do not form solid bicarbonate, but react with $\mathrm{N}_{2}$ to give nitrides.
<br/> <br/>$\begin{array}{l}
<br/> <br/>6 \mathrm{Li}+\mathrm{N}_{2} \stackrel{\Delta}{\rightarrow} 2 \mathrm{Li}_{3} \mathrm{~N} \\
<br/> <br/>3 \mathrm{Mg}+\mathrm{N}_{2} \stackrel{\Delta}{\rightarrow} \mathrm{Mg}_{3} \mathrm{~N}_{2}
<br/> <br/>\end{array}$</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
title:
two-equal-sides-of-an-isosceles-triangle-are-7-x-y-3-0-and-x-y-3-0-and-its-third-side-passes-through-the-point-1-10-the-equation-of-the-third-side-is

slug:
two-equal-sides-of-an-isosceles-triangle-are-7-x-y-3-0-and-x-y-3-0-and-its-third-side-passes-through-the-point-1-10-the-equation-of-the-third-side-is-45506

question:
<div class="question">Two equal sides of an isosceles triangle are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> and its third side passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>10</mn><mo>)</mo></math>. The equation of the third side is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>31</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>31</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>31</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>31</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>31</mn></math></span> </div>

answer:
<div class="solution"><p>The third side is parallel to a bisector of the angle between equal sides.</p><p>The bisectors are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>3</mn></mrow><msqrt><msup><mn>7</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mo>±</mo><mfrac><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mn>7</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>5</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>=</mo><mo>±</mo><mfrac><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow><msqrt><mn>2</mn></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>7</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>3</mn><mo>=</mo><mo>±</mo><mn>5</mn><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>3</mn><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>6</mn><mi>y</mi><mo>+</mo><mn>18</mn><mo>=</mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></p><p>Let the third side be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mi>k</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>L</mi></math></p><p>It passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>10</mn><mo>)</mo></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>k</mi><mo>=</mo><mn>31</mn><mo>,</mo><mo> </mo><mi>L</mi><mo>=</mo><mo>-</mo><mn>7</mn></math></p><p>Hence, required lines are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>31</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>7</mn></math>. </p></div>


tags: ['Mathematics', 'Sets and Relations', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7,4</span> </div>
source_db: MarksBatch2_P3.db
title:
two-finite-sets-have-m-and-n-elements-the-number-of-subsets-of-the-first-set-is-112-more-than-that-of-the-second-set-the-values-of-m-and-n-respectivel

slug:
two-finite-sets-have-m-and-n-elements-the-number-of-subsets-of-the-first-set-is-112-more-than-that-of-the-second-set-the-values-of-m-and-n-respectivel-34207

question:
<div class="question">Two finite sets have $m$ and $n$ elements. The number of subsets of the first set is 112 more than that of the second set. The values of $m$ and $n$ respectively are,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4,7</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">7,4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4,4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">7,7</span> </li> </ul>

answer:
<div class="solution">$2^{m}-2^{n}=112 \Rightarrow 2^{n}\left(2^{m-n}-1\right)=16.7$
<br/> <br/>$\quad \therefore \quad 2^{n}\left(2^{m-n}-1\right)=2^{4}\left(2^{3}-1\right)$
<br/> <br/>Comparing we get $n=4$ and $m-n=3$ $\Rightarrow n=4$ and $m=7$</div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.98</span> </div>
source_db: MarksBatch2_P3.db
title:
two-gases-occupy-two-containers-a-and-b-the-gas-in-a-of-volume-010-m-3-exerts-a-pressure-of-140-mpa-and-that-in-b-of-volume-015-m-3-exerts-a-pressure-

slug:
two-gases-occupy-two-containers-a-and-b-the-gas-in-a-of-volume-010-m-3-exerts-a-pressure-of-140-mpa-and-that-in-b-of-volume-015-m-3-exerts-a-pressure-83149

question:
<div class="question">Two gases occupy two containers $A$ and $B$ the gas in $\mathrm{A},$ of volume $0.10 \mathrm{~m}^{3},$ exerts a pressure of $1.40 \mathrm{MPa}$ and that in $\mathrm{B}$ of volume $0.15 \mathrm{~m}^{3}$ exerts a pressure $0.7 \mathrm{MPa}$. The two containers are united by a tube of negligible volume and the gases are allowed to intermingle. Then if the temperature remains constant, the final pressure in the container will be (in MPa)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.7</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">0.98</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1.4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">210</span> </li> </ul>

answer:
<div class="solution">We know that $\mathrm{P}_{\mathrm{A}} \mathrm{V}_{\mathrm{A}}=\mathrm{n}_{\mathrm{A}} \mathrm{RT}, \mathrm{P}_{\mathrm{B}} \mathrm{V}_{\mathrm{B}}=\mathrm{n}_{\mathrm{B}} \mathrm{RT}$
<br/> <br/>$\begin{array}{ll} &amp; \text { and } P_{f}\left(V_{A}+V_{B}\right)=\left(n_{A}+n_{B}\right) R T \\ &amp; P_{f}\left(V_{A}+V_{B}\right)=P_{A} V_{A}+P_{B} V_{B} \\ \therefore &amp; P_{f}=\left(\frac{P_{A} V_{A}+P_{B} V_{B}}{V_{A}+V_{B}}\right) \\ &amp; =\frac{1.4 \times 0.1+0.7 \times 0.15}{0.1+0.15} \mathrm{MPa}=0.98 \mathrm{MPa}\end{array}$</div>


tags: ['Physics', 'Capacitance', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
title:
two-identical-capacitors-have-the-same-capacitance-c-one-of-them-is-charged-to-potential-v-1-and-the-other-to-v-2-the-negative-ends-of-the-capacitors-

slug:
two-identical-capacitors-have-the-same-capacitance-c-one-of-them-is-charged-to-potential-v-1-and-the-other-to-v-2-the-negative-ends-of-the-capacitors-73545

question:
<div class="question">Two identical capacitors, have the same capacitance $\mathrm{C}$. One of them is charged to potential $\mathrm{V}_{1}$ and the other to $\mathrm{V}_{2}$. The negative ends of the capacitors are connected together. When the positive ends are also connected, the decrease in energy of the combined system is -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{4} \mathrm{C}\left(\mathrm{V}_{1}^{2}-\mathrm{V}_{2}^{2}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{4} C\left(V_{1}^{2}+V_{2}^{2}\right)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4} \mathrm{C}\left(\mathrm{V}_{1}-\mathrm{V}_{2}\right)^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{4} C\left(V_{1}+V_{2}\right)^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{4} \mathrm{C}\left(\mathrm{V}_{1}-\mathrm{V}_{2}\right)^{2}$</span> </div>

answer:
<div class="solution">Initial energy of combined system
<br/> <br/>$$
<br/> <br/>\mathrm{U}_{1}=\frac{1}{2} \mathrm{CV}_{1}^{2}+\frac{1}{2} \mathrm{CV}_{2}^{2}
<br/> <br/>$$
<br/> <br/>Final common potential, $\mathrm{V}=\frac{\mathrm{V}_{1}+\mathrm{V}_{2}}{2},$
<br/> <br/>Final energy of system,
<br/> <br/>$$
<br/> <br/>\mathrm{U}_{2}=2 \times \frac{1}{2} \mathrm{C}\left(\frac{\mathrm{V}_{1}+\mathrm{V}_{2}}{2}\right)^{2}
<br/> <br/>$$
<br/> <br/>Hence loss of energy $=\mathrm{U}_{1}-\mathrm{U}_{2}$
<br/> <br/>$$
<br/> <br/>=\frac{1}{4} \mathrm{C}\left(\mathrm{V}_{1}-\mathrm{V}_{2}\right)^{2}
<br/> <br/>$$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P3.db
title:
two-identical-charged-spheres-suspended-from-a-common-point-by-two-massless-strings-of-lengths-i-are-initially-at-a-distance-d-d-1-apart-because-of-th

slug:
two-identical-charged-spheres-suspended-from-a-common-point-by-two-massless-strings-of-lengths-i-are-initially-at-a-distance-d-d-1-apart-because-of-th-25926

question:
<div class="question">Two identical charged spheres suspended from a common point by two massless strings of lengths I, are initially at a distance d $(\mathrm{d}&lt;&lt;1)$ apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity $v$. Then $v$ varies as a function of the distance $\mathrm{x}$ between the spheres, as:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad v \propto x^{\frac{1}{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$v \propto x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$v \propto x^{-\frac{1}{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad v \propto x^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$v \propto x^{-\frac{1}{2}}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/DTRzdyMoussxQgg4C-oltdiGOAWvRE5VnSHJsRkUHPY.original.fullsize.png"/><br/> <br/> <br/>From figure $\tan \theta=\frac{\mathrm{F}_{\mathrm{e}}}{\mathrm{mg}} \simeq \theta$
<br/> <br/>$$
<br/> <br/>\frac{\mathrm{kq}^{2}}{\mathrm{x}^{2} \mathrm{mg}}=\frac{\mathrm{x}}{2 \ell}
<br/> <br/>$$
<br/> <br/>or $x^{3} \propto q^{2} \ldots$ (1)
<br/> <br/>or $x^{3 / 2} \propto q \ldots(2)$
<br/> <br/>Differentiating eq. (1) w.r.t. time
<br/> <br/>$3 \mathrm{x}^{2} \frac{\mathrm{dx}}{\mathrm{dt}} \propto 2 \mathrm{q} \frac{\mathrm{dq}}{\mathrm{dt}}$ but $\frac{\mathrm{dq}}{\mathrm{dt}}$ is constant
<br/> <br/>$$
<br/> <br/>\text { so } \mathrm{x}^{2}(\mathrm{v}) \propto \mathrm{q} \text { Replace } \mathrm{q} \text { from eq. }(2)
<br/> <br/>$$
<br/> <br/>$\mathrm{x}^{2}(\mathrm{v}) \propto \mathrm{x}^{3 / 2} \quad$ or $\quad \mathrm{v} \propto \mathrm{x}^{-1 / 2}$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
title:
two-identical-particles-each-of-mass-m-go-round-a-circle-of-radius-a-under-the-action-of-their-mutual-gravitational-attraction-the-angular-speed-of-ea-1

slug:
two-identical-particles-each-of-mass-m-go-round-a-circle-of-radius-a-under-the-action-of-their-mutual-gravitational-attraction-the-angular-speed-of-ea-1-19790

question:
<div class="question">Two identical particles each of mass ' $m$ ' go round a circle of radius $a$ under the action of their mutual gravitational attraction. The angular speed of each particle will be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{\frac{G m}{2 a^3}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{\frac{\mathrm{Gm}}{8 \mathrm{a}^3}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{\frac{G m}{4 a^3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{\frac{G m}{a^3}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{\frac{G m}{4 a^3}}$</span> </div>

answer:
<div class="solution">Radius of circle, $\mathrm{r}=\mathrm{a}$<br/>Centrifugal force, $F=m \omega^2 r$<br/>$\begin{aligned}<br/>&amp; \Rightarrow \frac{\mathrm{Gmm}}{(2 \mathrm{a})^2}=\mathrm{m} \omega^2 \mathrm{a} \\<br/>&amp; \left(\because \mathrm{F}=\frac{\mathrm{G} \mathrm{M} \mathrm{M}_2}{\mathrm{~d}}\right)<br/>\end{aligned}$<br/>Here $M_1=m ; M_2=m$ and $d=2 a$<br/>$\Rightarrow$ angular speed, $\omega=\sqrt{\frac{G m}{4 a^3}}$</div>


tags: ['Physics', 'Electromagnetic Induction', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
two-inductors-l-1-and-l-2-are-connected-in-parallel-and-a-time-varying-current-flows-as-shown-in-figure-the-ratio-ot-current-i-1-i-2-at-any-ime-t-is-1

slug:
two-inductors-l-1-and-l-2-are-connected-in-parallel-and-a-time-varying-current-flows-as-shown-in-figure-the-ratio-ot-current-i-1-i-2-at-any-ime-t-is-1-22015

question:
<div class="question"><p>Two inductors <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> are connected in parallel and a time varying current flows as shown in figure. The ratio ot current <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>i</mi><mn>1</mn></msub><msub><mi>i</mi><mn>2</mn></msub></mfrac></math> at any ime <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> Is<img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e395eb82-c1b7-4c9b-8cae-857766929c90-image.png" style="width: 350px; height: 174px;"/></p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>1</mn></msub></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>L</mi><mn>1</mn></msub><msub><mi>L</mi><mn>2</mn></msub></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msubsup><mi>L</mi><mn>2</mn><mn>2</mn></msubsup><msup><mfenced><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>+</mo><msub><mi>L</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msubsup><mi>L</mi><mn>1</mn><mn>2</mn></msubsup><msup><mfenced><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>+</mo><msub><mi>L</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>1</mn></msub></mfrac></math></span> </div>

answer:
<div class="solution"><p>As the inductors are in parallel, induced emf across the two inductors is the same, i.e.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub><mo>=</mo><msub><mi>e</mi><mn>2</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub><mfenced><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>1</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mfenced><mo>=</mo><msub><mi>L</mi><mn>2</mn></msub><mfenced><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>2</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mfenced></math></p><p>On integrating both sides, we get<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub><mo>∫</mo><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>1</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>L</mi><mn>2</mn></msub><mo>∫</mo><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>2</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub><msub><mi>i</mi><mn>1</mn></msub><mo>=</mo><msub><mi>L</mi><mn>2</mn></msub><msub><mi>i</mi><mn>2</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><msub><mi>i</mi><mn>1</mn></msub><msub><mi>i</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>1</mn></msub></mfrac></math></p></div>


tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
title:
two-inductors-l-1-and-l-2-are-connected-in-parallel-and-a-time-varying-current-flows-as-shown-in-figure-the-ratio-ot-current-i-1-i-2-at-any-ime-t-is

slug:
two-inductors-l-1-and-l-2-are-connected-in-parallel-and-a-time-varying-current-flows-as-shown-in-figure-the-ratio-ot-current-i-1-i-2-at-any-ime-t-is-97998

question:
<div class="question"><p>Two inductors <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> are connected in parallel and a time varying current flows as shown in figure. The ratio ot current <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>i</mi><mn>1</mn></msub><msub><mi>i</mi><mn>2</mn></msub></mfrac></math> at any ime <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> Is<img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e395eb82-c1b7-4c9b-8cae-857766929c90-image.png" style="width: 350px; height: 174px;"/></p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>1</mn></msub></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>L</mi><mn>1</mn></msub><msub><mi>L</mi><mn>2</mn></msub></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msubsup><mi>L</mi><mn>2</mn><mn>2</mn></msubsup><msup><mfenced><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>+</mo><msub><mi>L</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msubsup><mi>L</mi><mn>1</mn><mn>2</mn></msubsup><msup><mfenced><mrow><msub><mi>L</mi><mn>1</mn></msub><mo>+</mo><msub><mi>L</mi><mn>2</mn></msub></mrow></mfenced><mn>2</mn></msup></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>1</mn></msub></mfrac></math></span> </div>

answer:
<div class="solution"><p>As the inductors are in parallel, induced emf across the two inductors is the same, i.e.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub><mo>=</mo><msub><mi>e</mi><mn>2</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub><mfenced><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>1</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mfenced><mo>=</mo><msub><mi>L</mi><mn>2</mn></msub><mfenced><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>2</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mfenced></math></p><p>On integrating both sides, we get<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub><mo>∫</mo><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>1</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi>L</mi><mn>2</mn></msub><mo>∫</mo><mfrac><mrow><mi>d</mi><msub><mi>i</mi><mn>2</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub><msub><mi>i</mi><mn>1</mn></msub><mo>=</mo><msub><mi>L</mi><mn>2</mn></msub><msub><mi>i</mi><mn>2</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><msub><mi>i</mi><mn>1</mn></msub><msub><mi>i</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>L</mi><mn>2</mn></msub><msub><mi>L</mi><mn>1</mn></msub></mfrac></math></p></div>


title: two-lines-l-1-x-5-3-y-2-z-l-2-x-1-y-2-z-are-coplanar-then-can-take-value-s
slug: two-lines-l-1-x-5-3-y-2-z-l-2-x-1-y-2-z-are-coplanar-then-can-take-value-s-91269
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1,4,5</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">Two lines $L_{1}: x=5, \frac{y}{3-\alpha}=\frac{z}{-2}$
<br/> <br/>$\mathrm{L}_{2}: \mathrm{x}=\alpha, \frac{\mathrm{y}}{-1}=\frac{\mathrm{z}}{2-\alpha}$ are coplanar. Then, $\alpha \mathrm{can}$ take value (s)</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1,4,5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1,2,5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3,4,5</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2,4,5</span> </li> </ul>

answer:
<div class="solution">The equations of given lines can be written
<br/> <br/>as
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{L}_{1}: \mathrm{x}-5=\frac{\mathrm{y}}{3-\alpha}=\frac{\mathrm{z}}{-2} \\
<br/> <br/>\mathrm{~L}_{2}: \mathrm{x}-\alpha=\frac{\mathrm{y}}{-1}=\frac{\mathrm{z}}{2-\alpha}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Since, these lines are coplanar.
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\text { Therefore, }\left|\begin{array}{ccc}
<br/> <br/>5-\alpha &amp; 0-0 &amp; 0-0 \\
<br/> <br/>0 &amp; 3-\alpha &amp; -2 \\
<br/> <br/>0 &amp; -1 &amp; 2-\alpha
<br/> <br/>\end{array}\right|=0 \\
<br/> <br/>\Rightarrow \quad(5-\alpha)(3-\alpha)(2-\alpha)-2=0 \\
<br/> <br/>\Rightarrow \quad(5-\alpha)\left(6-3 \alpha-2 \alpha+\alpha^{2}-2\right]=0 \\
<br/> <br/>\Rightarrow \quad(5-\alpha)\left(\alpha^{2}-5 \alpha+4\right]=0 \\
<br/> <br/>\Rightarrow \quad(5-\alpha)(\alpha-1)(\alpha-4)=0 \\
<br/> <br/>\Rightarrow \quad \alpha=1,4,5
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
two-long-coaxial-and-conducting-cylinders-of-radius-a-and-b-are-separated-by-a-material-of-conductivity-and-a-constant-potential-difference-v-is-maint

slug:
two-long-coaxial-and-conducting-cylinders-of-radius-a-and-b-are-separated-by-a-material-of-conductivity-and-a-constant-potential-difference-v-is-maint-47675

question:
<div class="question">Two long coaxial and conducting cylinders of radius a and $\mathrm{b}$ are separated by a material of conductivity $\sigma$ and a constant potential difference $\mathrm{V}$ is maintained between them, by a battery. Then the current, per unit length of the cylinder flowing from one cylinder to the other is -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{4 \pi \sigma}{\ln (b / a)} V$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{4 \pi \sigma}{(b+a)} V$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{2 \pi \sigma}{\ln (b / a)} V$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2 \pi \sigma}{(b+a)} V$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2 \pi \sigma}{\ln (b / a)} V$</span> </div>

answer:
<div class="solution">$\mathrm{V}=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{E} \cdot \mathrm{d} \ell=\frac{\lambda}{2 \pi \varepsilon_{0} \mathrm{r}} \ln \left(\frac{\mathrm{b}}{\mathrm{a}}\right)$
<br/> <br/>Now, $I=\sigma \int \vec{E} \cdot \overrightarrow{d A}=\sigma \int \frac{\lambda}{2 \pi \varepsilon_{0} r} \cdot 2 \pi d r=\frac{\sigma \lambda}{\varepsilon_{0}}$
<br/> <br/>From (1)$: I=\frac{2 \sigma \pi \varepsilon_{0}}{\varepsilon_{0} \ln (b / a)}=\frac{2 \pi \sigma}{\ln (b / a)} v$</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
two-mercury-drops-each-of-radius-r-merge-to-form-a-bigger-drop-the-surface-energy-of-the-bigger-drop-if-t-is-the-surface-tension-is-1

slug:
two-mercury-drops-each-of-radius-r-merge-to-form-a-bigger-drop-the-surface-energy-of-the-bigger-drop-if-t-is-the-surface-tension-is-1-48928

question:
<div class="question">Two mercury drops (each of radius $\mathrm{r}$ ) merge to form a bigger drop. The surface energy of the bigger drop, if $\mathrm{T}$ is the surface tension, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2^{5 / 3} \pi r^{2} T$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \pi \mathrm{r}^{2} \mathrm{~T}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \pi^{2} T$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2^{8 / 3} \pi \mathrm{r}^{2} \mathrm{~T}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2^{8 / 3} \pi \mathrm{r}^{2} \mathrm{~T}$</span> </div>

answer:
<div class="solution">$\frac{4}{3} \pi R^{3}=2 \times \frac{4}{3} \pi r^{3} \Rightarrow \mathrm{R}=2^{1 / 3} \mathrm{r}$<br/>Surface energy of bigger drop,<br/>$\mathrm{E}=4 \pi R^{2} T=4 \times 2^{2 / 3} \pi r^{2} T=2^{8 / 3} \pi r^{2} T$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
title:
two-metal-plates-having-a-potential-difference-of-800-v-are-2-cm-apart-it-is-found-that-a-particle-of-mass-1-96-10-15-kg-remains-suspended-in-the-regi

slug:
two-metal-plates-having-a-potential-difference-of-800-v-are-2-cm-apart-it-is-found-that-a-particle-of-mass-1-96-10-15-kg-remains-suspended-in-the-regi-11859

question:
<div class="question">Two metal plates having a potential difference of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>800</mn><mo> </mo><mi mathvariant="normal">V</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>cm</mi></math> apart. It is found that a particle of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>96</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup><mo> </mo><mi>kg</mi></math> remains suspended in the region between the plates. The charge on the particle must be (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo></math> elementary charge)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>e</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>e</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi>e</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mi>e</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>e</mi></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>800</mn><mo> </mo><mi mathvariant="normal">V</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mfrac><mi>V</mi><mi>d</mi></mfrac><mo>=</mo><mfrac><mn>800</mn><mrow><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo> </mo><mi mathvariant="normal">V</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo> </mo><mo> </mo><mi mathvariant="normal">V</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>At equilibrium electrostatic force must be equal to force due to weight.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mi>E</mi><mo>=</mo><mi>m</mi><mi>g</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>q</mi><mo>=</mo><mfrac><mrow><mi>m</mi><mi>g</mi></mrow><mi>E</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>96</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup><mo>×</mo><mn>9</mn><mo>.</mo><mn>8</mn></mrow><mrow><mn>4</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup></mrow></mfrac></math> coulomb</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>96</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup><mo>×</mo><mn>9</mn><mo>.</mo><mn>8</mn></mrow><mrow><mn>4</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo>×</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>19</mn></mrow></msup></mrow></mfrac></math> elementary charges</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo> </mo><mi>e</mi></math></p></div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
title:
two-metallic-spheres-of-radii-1-cm-and-3-cm-are-given-charges-of-1-1-0-2-c-and-5-1-0-2-c-respectively-if-these-are-connected-by-a-conducting-wire-the-

slug:
two-metallic-spheres-of-radii-1-cm-and-3-cm-are-given-charges-of-1-1-0-2-c-and-5-1-0-2-c-respectively-if-these-are-connected-by-a-conducting-wire-the-34400

question:
<div class="question">Two metallic spheres of radii $1 \mathrm{~cm}$ and $3 \mathrm{~cm}$ are given charges of $-1 \times 10^{-2} \mathrm{C}$ and $5 \times 10^{-2} \mathrm{C}$, respectively. If these are connected by a conducting wire, the final charge on the bigger sphere is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \times 10^{-2} \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3 \times 10^{-2} C$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 \times 10^{-2} C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1 \times 10^{-2} C$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \times 10^{-2} C$</span> </div>

answer:
<div class="solution">$\quad \frac{k x}{1 \mathrm{~cm}}=\frac{k(Q-x)}{3 \mathrm{~cm}}$<br/>$3 x=Q-x \Rightarrow 4 x=Q$<br/>$x=\frac{Q}{4}=\frac{4 \times 10^{-2}}{4} C=1 \times 10^{-2}$<br/>$Q^{\prime}=\mathrm{Q}-x=3 \times 10^{-2} \mathrm{C}$</div>


